Special Issue No. – 12, August 2025

Abstract:

In this research, we provide novel methods for analyzing some models of PDEs that occur in a wide range of scientific and engineering applications. Adomian polynomials have been utilized for this purpose. The simplicity and accuracy of the suggested Integrated technique are confirmed by combining the Adomian decomposition method with the conventional Double Elzaki transform. An experimental study has been conducted. To illustrate the efficiency of the proposed scheme, the Rangaig transform-based Homotopy analysis method is used for the comparison study of the solutions.

Keywords:

Adomian decomposition method,Double Elzaki transform,Benjamin-Bona-Mahony equations,KdV equations,Linear Schrodinger equations,Test examples,

Refference:

I. Abbasbandy, S. “The Application of Homotopy Analysis Method to Solve a Generalized Hirota-Satsuma Coupled KdV Equation.” Physics Letters A, vol. 7, 2007, pp. 478-83.
II. Ahmed, S. “Application of Sumudu Decomposition Method for Solving Burger’s Equation.” Advances in Theoretical and Applied Mathematics, vol. 9, 2014, pp. 23-26.
III. Alderremy, A. A., and T. M. Elzakib. “On the New Double Integral Transform for Solving Singular System of Hyperbolic Equations.” Journal of Nonlinear Sciences and Applications, vol. 11, 2018, pp. 1207-14.
IV. Dehghan, M., et al. “Solving Nonlinear Partial Differential Equation Using the Homotopy Analysis Method.” Numerical Methods for Partial Differential Equations, vol. 26, no. 2, 2010, pp. 448-63.
V. Elzaki, Tarig. “The New Integral Transform Elzaki Transform.” Global Journal of Pure and Applied Mathematics, vol. 7, no. 1, 2011, pp. 57-64.
VI. Elzaki, Tarig, and Salih M. Elzaki. “On the Connections between Laplace and Elzaki Transforms.” Advances in Theoretical and Applied Mathematics, vol. 6, no. 1, 2011, pp. 1-10.
VII. —. “On the Elzaki Transform and Ordinary Differential Equation with Variable Coefficients.” Advances in Theoretical and Applied Mathematics, vol. 6, no. 1, 2011, pp. 41-46.
VIII. Elzaki, Tarig M., and Eman M. A. Hilal. “Solution of Linear and Non-linear Partial Differential Equations Using Mixture of Elzaki Transform and the Projected Differential Transform Method.” Mathematical Theory and Modeling, vol. 2, no. 4, 2012.
IX. Elzaki, Tarig M., et al. “Elzaki and Sumudu Transforms for Solving Some Differential Equation.” Global Journal of Pure and Applied Mathematics, vol. 8, no. 2, 2012, pp. 167-73.
X. Hassaballa, Abaker A., and Yagoub A. Salih. “On Double Elzaki Transform and Double Laplace Transform.” IOSR Journal of Mathematics, vol. 11, no. 1, 2015, pp. 35-41.
XI. Rangaig, N. A., et al. “On Another Type of Transform Called Rangaig Transform.” International Journal of Partial Differential Equations and Applications, vol. 5, no. 1, 2017, pp. 42-48.
XII. Ziane, D., and M. H. Cherif. “The Homotopy Analysis Rangaig Transform Method for Nonlinear Partial Differential Equations.” Journal of Applied Mathematics and Computational Mechanics, vol. 21, no. 2, 2022, pp. 111-22.

View | Download

Abstract:

This study demonstrates a semi-analytic method for solving two-dimensional wave equations that arise in several scientific and engineering fields by combining the Laplace Transform with a corrected variational iteration technique. A few numerical examples are provided to illustrate the correctness of the suggested method.

Keywords:

Variational Iterative method,Laplace Transform,Numerical Examples,Two-dimensional wave equation,

Refference:

I. Arife, A. S., and A. Yildirim. “New Modified Variational Iteration Transform Method (MVITM) for Solving Eighth-Order Boundary Value Problems in One Step.” World Applied Sciences Journal, vol. 13, no. 10, 2011, pp. 2186–2190.
II. Biazar, J., and M. Eslami. “A New Technique for Nonlinear Two Dimensional Wave Equations.” Scientia Iranica, vol. 20, no. 2, 2013, pp. 359–363.
III. Elzaki, T. M. “Solution of Nonlinear Partial Differential Equations by New Laplace Variational Iteration Method.” Differential Equations: Theory and Current Research, 2018.
IV. Gr. Ixaru, L. “Operations on Oscillatory Functions.” Computer Physics Communication, vol. 105, 1997, pp. 1–9.
V. Hajj, F. Y. “Solution of the Schrodinger Equation in Two and Three Dimensions.” J. Phys. B At. Mol. Physics, vol. 18, 1985, pp. 1–11.
VI. Hammouch, Z., and T. Mekkaoui. “A Laplace-Variational Iteration Method for Solving the Homogeneous Smoluchowski Coagulation Equation.” Applied Mathematical Sciences, vol. 6, no. 18, 2012, pp. 879–886.
VII. He, J. H. “An Approximation to Solution of Space and Time Fractional Telegraph Equations by the Variational Iteration Method.” Mathematical Problems in Engineering, vol. 2012, 2012, pp. 1–2.
VIII. He, J. H. “Variational Iteration Method for Autonomous Ordinary Differential Systems.” Applied Mathematics and Computation, vol. 114, no. 2–3, 2000, pp. 115–123.
IX. He, J. H. “Variational Iteration Method for Delay Differential Equations.” Communications in Nonlinear Science and Numerical Simulation, vol. 2, no. 4, 1997, pp. 235–236.
X. He, J. H. “Variational Iteration Method—A Kind of Non-Linear Analytical Technique: Some Examples.” International Journal of Non-Linear Mechanics, vol. 34, no. 4, 1999, pp. 699–708.
XI. Hesameddini, E., and H. Latifizadeh. “Reconstruction of Variational Iteration Algorithms Using the Laplace Transform.” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 11–12, 2009, pp. 1377–1382.
XII. Khuri, S. A., and A. Sayfy. “A Laplace Variational Iteration Strategy for the Solution of Differential Equations.” Applied Mathematics Letters, vol. 25, no. 12, 2012, pp. 2298–2305.
XIII. Levy, M. “Parabolic Equation Method for Electro-Magnetic Wave Propagation.” IEEE, 2000.
XIV. Macig, A., and J. Wauer. “Solution of Two Dimensional Wave Equation by Using Wave Polynomial.” Journal of Engineering Mathematics, vol. 51, 2005, pp. 339–350.
XV. Shah, R., H. Khan, D. Baleanu, P. Kumam, and M. Arif. “A Semi-Analytical Method for Solving Family of Kuramoto-Sivashinsky Equations.” Journal of Taibah University for Sciences, vol. 14, no. 1, 2020, pp. 402–411.
XVI. Singh, G., and I. Singh. “New Laplace Variational Iterative Method for Solving 3D Schrödinger Equations.” Journal of Mathematical and Computational Science, vol. 10, no. 5, 2020, pp. 2015–2024.
XVII. Singh, I., and S. Kumar. “Wavelet Methods for Solving Three-Dimensional Partial Differential Equations.” Mathematical Sciences, vol. 11, 2017, pp. 145–154.
XVIII. Tappert, F. D. “The Parabolic Approximation Method.” In: Keller, J. B., and J. S. Papadakis (Eds.), Wave Propagation and Underwater Acoustics. Lecture Notes in Physics, Springer, Berlin, vol. 70, 1977, pp. 224–287.
XIX. Ullah, H., S. Islam, L. C. C. Dennis, T. N. Abdulhameed, I. Khan, and M. Fiza. “Approximate Solution of Two Dimensional Wave Equations by Optimal Homotopy Asymptotic Method.” Mathematical Problems in Engineering, 2015, pp. 1–7.
XX. Wu, G. C. “Variational Iteration Method for Solving the Time-Fractional Diffusion Equations in Porous Medium.” Chinese Physics B, vol. 21, no. 12, 2

View | Download

Abstract:

The Laplace Transform method and variational iterative approach are combined to create a new semi-analytical methodology that is used in this research to solve one-dimensional heat equations. To illustrate the effectiveness and precision of the suggested approach, numerical results are provided.

Keywords:

Variational iterative method,one-dimensional Heat equation,Numerical examples,Laplace transform,

Refference:

1. Arife, A. S., and A. Yildirim. “New Modified Variational Iteration Transform Method (MVITM) for Solving Eighth-Order Boundary Value Problems in One Step.” World Applied Sciences Journal, vol. 13, no. 10, 2011, pp. 2186–2190.
2. Douglas, J., and D. W. Peaceman. “Numerical Solution of Two Dimensional Heat Flow Problems.” AIChE Journal, vol. 1, no. 4, 1955, pp. 505–512.
3. Hammouch, Z., and T. Mekkaoui. “A Laplace-Variational Iteration Method for Solving the Homogeneous Smoluchowski Coagulation Equation.” Applied Mathematical Sciences, vol. 6, no. 18, 2012, pp. 879–886.
4. He, J. H. “Variational Iteration Method—A Kind of Non-Linear Analytical Technique: Some Examples.” International Journal of Non-Linear Mechanics*, vol. 34, no. 4, 1999, pp. 699–708.
5. Hesameddini, E., and H. Latifizadeh. “Reconstruction of Variational Iteration Algorithms Using the Laplace Transform.” *International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 11-12, 2009, pp. 1377–1382.
6. Khuri, S. A., and A. Sayfy. “A Laplace Variational Iteration Strategy for the Solution of Differential Equations.” Applied Mathematics Letters, vol. 25, no. 12, 2012, pp. 2298–2305.
7. Kouatchou, J. “Finite Difference and Collocation Methods for Solution of Two Dimensional Heat Equation.” Numerical Methods for Partial Differential Equations, vol. 17, no. 1, 2001, pp. 54–63.
8. Luga, T., T. Aboiyar, and S. O. Adee. “Radial Basis Function Methods for Approximating the Two Dimensional Heat Equations.” International Journal of Engineering Applied Sciences and Technology, vol. 4, no. 2, 2019, pp. 7–15.
9. Murphy, C. P., and D. J. Evans. “Chebyshev Series Solution of Two Dimensional Heat Equation.” Mathematics and Computers in Simulation, vol. 23, no. 2, 1981, pp. 157–162.
10. Shah, R., H. Khan, D. Baleanu, P. Kumam, and M. Arif. “A Semi-Analytical Method for Solving Family of Kuramoto-Sivashinsky Equations.” Journal of Taibah University for Sciences, vol. 14, no. 1, 2020, pp. 402–411.
11. Singh, G., and I. Singh. “New Laplace Variational Iterative Method for Solving 3D Schrödinger Equations.” Journal of Mathematical and Computational Science, vol. 10, no. 5, 2020, pp. 2015–2024.
12. Wu, G. C. “Variational Iteration Method for Solving the Time-Fractional Diffusion Equations in Porous Medium.” Chinese Physics B, vol. 21, no. 12, 2012.

View | Download

Abstract:

Hepatitis B remains a serious global health concern, affecting approximately one-third of the world’s population and causing nearly one million deaths annually. The SEI_C I_A R model that distinguishes between acutely and chronically infected individuals becomes a significant addition to public health research about Hepatitis B virus transmission. This study provides rigorous insights using fixed-point theory with generalised Hyers-Ulam stability criteria to produce thorough results about solution existence, uniqueness, and stability. The model demonstrates through visualisation using the RK-5 method that proper population control measures, such as vaccination systems, transmission rate, lead populations toward the eradication of disease states. This research both enhances mathematical epidemiology and supports worldwide hepatitis B elimination programs.

Keywords:

Hepatitis B Virus,Banach Contraction Principle,Fixed Point Approach,Picard Theorem,Generalised Hyers-Ulam stability,

Refference:

I. Ansari, Q. H. Metric Spaces: Including Fixed Point Theory and Set-Valued Maps. Alpha Science International, 2010.
II. Butsashvili, M., Tsertsvadze, T., McNutt, L., Kamkamidze, G., Gvetadze, R., and Badridze, N. “Prevalence of Hepatitis B, Hepatitis C, Syphilis and HIV in Georgian Blood Donors.” European Journal of Epidemiology, vol. 17, 2001, pp. 693–695. 10.1023/A:1020026117551.
III. Desta, B. S., and Koya, P. R. “Modified Model and Stability Analysis of the Spread of Hepatitis B Virus Disease.” American Journal of Applied Mathematics, vol. 7, 2019, pp. 13.
IV. Dienstag, J. L. “Hepatitis B Virus Infection.” New England Journal of Medicine, vol. 359, 2008, pp. 1486–1500. 10.1056/NEJMra0801644.
V. European Centre for Disease Prevention and Control (ECDC). Evidence, Prevention of Hepatitis B and C in the EU/EEA. 2024. https://www.ecdc.europa.eu/en/publications-data/prevention-hepatitis-b-and-c-eueea-2024.
VI. Garg, P., and Chauhan, S. S. “Stability Analysis of a Solution for the Fractional-Order Model on Rabies Transmission Dynamics Using a Fixed-Point Approach.” Mathematical Methods in the Applied Sciences, 2024, pp. 1–12. 10.1002/mma.10150.
VII. Hussain, A., Baleanu, D., and Adeel, M. “Existence of Solution and Stability for the Fractional Order Novel Coronavirus (nCoV-2019) Model.” Advances in Difference Equations, vol. 2020, 2020, pp. 1–9. 10.1186/s13662-020-02730-9.
VIII. Khan, N. “Metric Space, Applications and Its Properties.” International Journal of Scientific Research and Reviews, vol. 8, 2019, pp. 5.
IX. Kutbi, M. A., and Sintunavarat, W. “Ulam-Hyers Stability and Well-Posedness of Fixed Point Problems for α-λ-Contraction Mapping in Metric Spaces.” Abstract and Applied Analysis, vol. 2014, 2014, pp. 1–6. 10.1155/2014/589763.
X. McMahon, B. J. “The Natural History of Chronic Hepatitis B Virus Infection.” Hepatology, vol. 49, 2009, pp. S45–S55. 10.1002/hep.22898.
XI. Minggi, I., Ramadhan, N. R., and Side, S. “The Accuracy Comparison of the RK-4 and RK-5 Method of SEIR Model for Tuberculosis Cases in South Sulawesi.” Journal of Physics: Conference Series, vol. 1918, no. 4, 2021. 10.1088/1742-6596/1918/4/042020.
XII. Moneim, I., and Khalil, H. “Modelling and Simulation of the Spread of HBV Disease with Infectious Latent.” Applied Mathematics, vol. 6, 2015, pp. 745. 10.4236/am.2015.67070.
XIII. Nana-Kyere, S., Ackora-Prah, J., Okyere, E., Marmah, S., and Afram, T. “Hepatitis B Optimal Control Model with Vertical Transmission.” Applied Mathematics, vol. 7, 2017, pp. 5–13.
XIV. Phiangsungnoen, S., Sintunavarat, W., and Kumam, P. “Fixed Point Results, Generalized Ulam-Hyers Stability and Well-Posedness via α-Admissible Mappings in B-Metric Spaces.” Fixed Point Theory and Applications, 2014, pp. 1–17. 10.1186/1687-1812-2014-3.
XV. Seto, W.-K., Lo, Y.-R., Pawlotsky, J.-M., and Yuen, M.-F. “Chronic Hepatitis B Virus Infection.” The Lancet, vol. 392, 2018, pp. 2313–2324. 10.1016/S0140-6736(18)31865-8.
XVI. Singh, R., and Aggarwal, J. “Introduction to Metric Spaces.” 2016, pp. 1–50.

View | Download

Abstract:

In this paper, we propose a refined mathematical framework-termed the ‘ESIS model’, to address key limitations found in the classical SSEIR model of information propagation. Since the SSEIR model offers a foundational approach to capturing the dynamics of information spread, it falls short in representing scenarios where information circulates or stays active in a population over time. To overcome this, the ESIS model introduces a modified structure with additional compartments that more accurately represent the real-world flow of information. We develop its corresponding system of dynamic differential equations and offer a thorough state transition diagram to illustrate the behavior of individuals across different stages of information exposure. To assess the performance of the ESIS model, we simulate and compare it against the SSEIR framework through graphical analysis. The results indicate that the ESIS model enables more sustained and realistic propagation, making it a more effective tool for studying long-term influence in social networks and other information-driven systems.

Keywords:

Information,model,SSEIR,active,hypergraph,Social,Susceptible,

Refference:

I. Al-Oraiqat, Anas M., et al. “Modeling strategies for information influence dissemination in social networks.” Journal of Ambient Intelligence and Humanized Computing 13.5 (2022): 2463-2477. 10.1007/s12652-021-03364-w
II. Dhol, B. S., Gonder, S. S. C., & Kumar, N. “Information and Communication Technology-Based Math’s Education: A Systematic Review.” 2023 International Conference on Advancement in Computation & Computer Technologies (InCACCT). IEEE, (2023): pp. 618-622 10.1109/incacct57535.2023.10141689
III. Goffman, William, and V. A. Newill. “Generalization of Epidemic Theory.” Nature, vol. 204, no. 4955, 1964, pp. 225–228. 10.1038/204225a0
IV. Singh Chauhan, Surjeet, Shalni Chandra, and Prachi Garg. “Novel Insights Into Fixed‐Point Results on Graphical Cone b b‐Metric Space With Some Application to the System of Boundary Value Problems.” Mathematical Methods in the Applied Sciences (2025). 10.1002/mma.11151
V. HU, F., LI, F., & ZHAO, H. “The research on scale-free characteristics of hypernetworks.” Scientia Sinica Physica, Mechanica & Astronomica 47.6 (2017): 060501. 10.1360/sspma2016-00426
VI. Iamnitchi, Adriana, et al. “Modeling information diffusion in social media: data-driven observations.” Frontiers in Big Data 6 (2023): 1135191. 10.3389/fdata.2023.1135191
VII. Li, Li, et al. “Information cascades blocking through influential nodes identification on social networks.” Journal of Ambient Intelligence and Humanized Computing 14.6 (2023): 7519-7530.
10.1007/s12652-022-04456-x
VIII. Li, Mei, et al. “A survey on information diffusion in online social networks: Models and methods.” Information 8.4 (2017): 118. 10.3390/info8040118
IX. Liu, Yun, et al. “SHIR competitive information diffusion model for online social media.” Physica A: Statistical Mechanics and its Applications 461 (2016): 543-553. 10.1016/j.physa.2016.06.080
X. Rozemberczki, Benedek, Carl Allen, and Rik Sarkar. “Multi-scale attributed node embedding.” Journal of Complex Networks 9.2 (2021): cnab014. 10.1093/comnet/cnab014
XI. Seidman, Stephen B. “Structures induced by collections of subsets: A hypergraph approach.” Mathematical Social Sciences 1.4 (1981): 381-396. 10.1016/0165-4896(81)90016-0
XII. Tong, Qiujuan, et al. “The fractional SEIRS epidemic model for information dissemination in social networks.” Advances in Natural Computation, Fuzzy Systems and Knowledge Discovery: Volume 2. Springer International Publishing, 2020. 10.1007/978-3-030-32591-6_30
XIII. Vasilyeva, Ekaterina, et al. “Distances in higher-order networks and the metric structure of hypergraphs.” Entropy 25.6 (2023): 923. 10.3390/e25060923
XIV. Xiao, Hai-Bing, et al. “Information propagation in hypergraph-based social networks.” Entropy 26.11 (2024): 957. 10.3390/e26110957
XV. Wang, Ruiheng, et al. “User identity linkage across social networks by heterogeneous graph attention network modeling.” Applied Sciences 10.16 (2020): 5478. 10.3390/app10165478
XVI. Wang, Qiyao, et al. “ESIS: Emotion-based spreader–ignorant–stifler model for information diffusion.” Knowledge-based systems 81 (2015): 46-55. 10.1016/j.knosys.2015.02.006
XVII. Zhang, Chuangchuang, et al. “Hypergraph-Based Influence Maximization in Online Social Networks.” Mathematics 12.17 (2024): 2769. 10.3390/math12172769
XVIII. Zhu, Nafei, et al. “Modeling the dissemination of privacy information in online social networks.” Transactions on Emerging Telecommunications Technologies 35.6 (2024): e4989. 10.1002/ett.4989
XIX. Zou, Xingzhu, et al. “Information Diffusion Prediction Based on Deep Attention in Heterogeneous Networks.” International Conference on Spatial Data and Intelligence. Cham: Springer Nature Switzerland, (2022): pp. 99-112. 10.1007/978-3-031-24521-3_8

View | Download

Abstract:

To differentiate itself from other integral transformations, this article presents a novel integral transformation known as the A (or Aman) transform. This transformation was inspired by a thorough study of the effectiveness of the Laplace and Sumudu transforms, specifically with regard to fractional differential equations. Applying these transformations might occasionally make processing their inverse transform challenging. This concept encourages us to reconsider and put in more effort to develop fresh, essential transformations that will make difficult problems easier to tackle. The proposed transformation has been successfully used to solve the Riemann-Liouville and Caputo FDE analytically. The outcomes of using this new approach are in perfect harmony with those of using contemporary methods. This demonstrates the A transform's dependability and efficiency in the analytical resolution of complex mathematical situations

Keywords:

Laplace Transform,Sumudu Transform,Fractional Differential Equations,Riemann-Liouville's Fractional Differential Equations,

Refference:

I. Abdel Rahim Mahgoub, Mohand M. “The New Integral Transform Mahgoub Transform.” Advances in Theoretical and Applied Mathematics, vol. 11, no. 4, 2016, pp. 391–398. https://www.ripublication.com/atam16/atamv11n4_07.pdf
II. Abdel Rahim Mahgoub, Mohand M. “The New Integral Transform Mohand Transform.” Advances in Theoretical and Applied Mathematics, vol. 12, no. 2, 2017, pp. 113–120. https://www.ripublication.com/atam17/atamv12n2_07.pdf
III. Abdel Rahim Mahgoub, Mohand M. “The New Integral Transform Sawi Transform.” Advances in Theoretical and Applied Mathematics, vol. 14, no. 1, 2019, pp. 81–87. https://www.ripublication.com/atam19/atamv14n1_05.pdf
IV. Aboodh, Khalid Suliman. “The New Integral Transform Aboodh Transform.” Global Journal of Pure and Applied Mathematics, vol. 9, no. 1, 2013, pp. 35–43. https://www.ripublication.com/gjpamv7/gjpamv9n1_04.pdf
V. Belgacem, Rachid, Dumitru Baleanu, and Ahmed Bokhari. “Shehu Transform and Applications to Caputo-Fractional Differential Equations.” International Journal of Analysis and Applications, vol. 17, 2019, no. 6, pp. 917–927. 10.28924/2291-8639-17-2019-917VI.
VI. Bracewell, Ron, and Peter B. Kahn. “The Fourier Transform and Its Applications.” American Journal of Physics, vol. 34, no. 8, 1966, pp. 712–712. 10.1119/1.1973431.
VII. Bulut, H., H. M. Baskonus, and F. B. M. Belgacem. “The Analytical Solution of Some Fractional Ordinary Differential Equations by the Sumudu Transform Method.” Abstract and Applied Analysis, vol. 4, 2013, pp. 1–6. 10.1155/2013/203875
VIII. Debnath, Lokenath, and Dambaru Bhatta. “Integral Transforms and Their Applications. Chapman and Hall/CRC.” Taylor and Francis Group, 2007. 10.1201/b17670
IX. Elzaki, Tarig M. “The New Integral Transform Elzaki Transform.” Global Journal of Pure and Applied Mathematics, vol. 7, no. 1, 2011, pp. 57–64.
X. Elzaki, Tarig M., and Mourad Chamekh. “Solving Nonlinear Fractional Differential Equations Using a New Decomposition Method.” Universal Journal of Applied Mathematics and Computation, vol. 6, 2018, pp. 27–35.
XI. Erdélyi, A., et al. Tables of Integral Transforms. Vol. 1, McGraw-Hill, 1954.
XII. Goswami, Pranay, and Rubayyi T. Alqahtani. “Solutions of Fractional Differential Equations by Sumudu Transform and Variational Iteration Method.” Journal of Nonlinear Sciences and Applications, 2016. 10.22436/jnsa.009.04.48
XIII. Hailat, Ibrahim, Zarita Zainuddin, and Amirah Azmi. “New Approach of Modifying Laplace Transform Variational Iteration Method to Solve Fourth-Order Fractional Integro-Differential Equations.” Authorea, 30 Jan. 2024. 10.22541/au.170665886.63054022/v1
XIV. Liang, Song, Ranchao Wu, and Liping Chen. “Laplace Transform of Fractional Order Differential Equations.” Electronic Journal of Differential Equations, no. 139, 2015, pp. 1–15. http://ejde.math.txstate.edu/Volumes/2015/139/liang.pdf
XV. Miller, Kenneth S., and Bertram Ross. “An Introduction to the Fractional Calculus and Fractional Differential Equations.” John Wiley and Sons, 1993.
XVI. Moazzam, A., and Z. I. Muhammad. “A New Integral Transform ‘Ali and Zafar’ Transformation and Its Application in Nuclear Physics.” Proceedings of the 19th International Conference on Statistical Sciences, vol. 36, 2022, pp. 177–182. https://www.researchgate.net/publication/361409149.
XVII. Podlubny, Igor. Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198, Academic Press, January 15, 1999. URL: https://archive.org/details/fractionaldiffer00podl_878.
XVIII. Schi, J. L. “The Laplace Transform: Theory and Applications.” Springer, 1999.
XIX. Singh, Amandeep, et al. “Solving Nonlinear Coupled Fractional Partial Differential Equations by ZZ Transform and Adomian Polynomials.” Journal of Mechanics of Continua and Mathematical Sciences, Special Issue no. 11, May 2024, pp. 1–17. 10.26782/jmcms.spl.11/2024.05.00001
XX. Singh, Jagdev, Devendra Kumar, and A. Klman. “Homotopy Perturbation Method for Fractional Gas Dynamics Equation Using Sumudu Transform.” Abstract and Applied Analysis, 2013. 10.1155/2013/934060
XXI. Sontakke, Bhausaheb R., and Rajashri Pandit. “Convergence Analysis and Approximate Solution of Fractional Differential Equations.” Malaya Journal of Matematik, vol. 7, no. 2, 2019, pp. 338–344. 10.26637/MJM0702/0029
XXII. Tuluce Demiray, Seyma, Hasan Bulut, and Fethi Bin Muhammad Belgacem. “Sumudu Transform Method for Analytical Solutions of Fractional Type Ordinary Differential Equations.” Mathematical Problems in Engineering, 2015, Article ID 131690. 10.1155/2015/131690
XXIII. Watugala, G. K. “Sumudu Transform—An Integral Transform to Solve Differential Equations and Control Engineering Problems.” International Journal of Mathematical Education in Science and Technology, vol. 24, no. 1, 1993, pp. 35–43. 10.1080/0020739930240105
XXIV. Yang, Xiao-Jun. Local Fractional Functional Analysis and Its Applications. Asian Academic Publisher, 2011.
XXV. Zhao, Weidong, and Shehu Maitama. “New Integral Transform: Shehu Transform—A Generalization of Sumudu and Laplace Transform for Solving Differential Equations.” International Journal of Analysis and Applications, vol. 17, no. 2, 2019, pp. 167–190.
XXVI. Zhao, Weidong, and Shehu M. “Homotopy Perturbation Shehu Transform Method for Solving Fractional Models Arising in Applied Science.” Journal of Applied Mathematics and Computational Mechanics, vol. 20, no. 1, 2021, pp. 71–82.
XXVII. Yang, Yi, and Haiyan Henry Zhang. Fractional Calculus with its Applications in Engineering and Technology. Morgan & Claypool (Springer Nature Imprint), 2019. https://link.springer.com/book/10.1007/978-3-031-79625-8

View | Download

Abstract:

The proposed model addresses the limitations of existing analytical methods, such as equilibrium and lower-bound approaches, by considering height reduction, relative density increments, and varying die-workpiece interactions simultaneously. The model’s predictions align closely with experimental results, demonstrating its reliability and accuracy. This research paves the way for the development of industrial-grade software tools, offering manufacturers a competitive edge by improving process predictability and reducing dependence on computationally intensive methods like the Finite Element Approach.

Keywords:

Sinter Forging,Analytical Modeling,Upper Bound Dynamics,Aspect ratio,

Refference:

I. Agrawal M., Jha A. K., Kumar S., “High-speed forging of hollow metal powder preforms”, Inst. Engrs. (I) J., 80 (1999),8.
II. Avitzur B., “Metal Forming Processes and Analysis”, McGraw Hill, New York, 1968.
III. Cost savings win the day for PM parts, Metal Powder Report, Vol. 56, Issue 7-8, (2001), 10-14.
IV. Jain Shrikant, Ranjan R. K. and Kumar Surender, “Fracturing and Deformation Characteristics of Aluminium Preform during Cold Forging at Low Strain Rates”. Int. J. of Scientific Engineering and Technology, 4(3) (2015),182-186. 10.17950/ijset/v4s3/314
V. Jha A. K and Kumar S., “Analysis of Axisymmetric Cold Processing of Metal Powder Preforms”, Journal of the Institution of Engineers (India), 65, (1985), 169.
VI. Jha A.K., Kumar S., “Dynamic effects during a high-speed sinter-forging process”, International Journal of Machine Tools and Manufacture, Vol. 36, Issue 10, (1996), 1109-1122, ISSN 0890-6955. 10.1016/0890-6955(95)00122-0
VII. Jha A. K., Kumar S., “Investigations into the high-speed forging of sintered copper powder strips”, Journal of Materials Processing Technology,71(3), (1997), 394-401. 10.1016/S0924-0136(97)00104-0
VIII. Jha A. K., Kumar S., “Compatibility of sintered materials during cold forging”, International Journal of Materials and Product Technology’ 9, Issue 4-6, (2004), 281-299. 10.1504/IJMPT.1994.036423
IX. Jones P. K., “The technical and economic advantage of powder forged products”, Powder Metallurgy, Vol. 13, Issue 26, (1970), 114-129.
X. Kumar Parveen, Ranjan R. K., Kumar Rajive, “Mechanics of deformation during open die forging of sintered preform: Comparative study by equilibrium and upper bound methods”, ARPN Journal of Engineering and Applied Sciences, 6(6), (2011),83–93. https://arpnjournals.com/jeas/research_papers/rp_2011/jeas_0611_515.pdf
XI. Kumar Parveen, Ranjan R. K., Kumar Rajive, “Investigations of an axisymmetric compound flow behavior of sintered preform: An upper bound approach”, International Journal of Pure and Applied Mathematics, Vol. 81, Issue 5, (2012), 671-691. https://www.ijpam.eu/contents/2012-81-5/2/2.pdf
XII. Kumar Parveen, Ranjan R. K., Kumar Rajive, “Mathematical modelling of forging of sintered preform: Comparative study of open and closed die”, International Journal of Pure and Applied Mathematics, Vol. 82, Issue 2, (2012), 179-188. https://www.ijpam.eu/contents/2013-82-2/2/2.pdf
XIII. Singh S., Jha A. K., “Sintered preforms adds better value to aerospace components”, Journal of Aerospace Engineering, I. E. (I), 82, (2001), 1-6.
XIV. Singh Saranjit, Jha A.K., “Analysis of dynamic effects during high-speed forging of sintered preforms”, Journal of Materials Processing Technology, Volume 112, Issue 1, 2001, Pages 53-62, ISSN 0924-0136 10.1016/S0924-0136(00)00898-0
XV. Singh S., Jha A. K., “An energy analysis during forging of sintered truncated conical preform at high-speed”, Tamkang J. of Science and Engineering,7,(2004),227-236. http://jase.tku.edu.tw/articles/jase-200412-7-4-05
XVI. Singh Saranjit & Jha A. K. & Kumar Suhas. “Dynamic effects during sinter forging of axi-symmetric hollow disc preforms”, International Journal of Machine Tools and Manufacture, Vol. 47, Issue 7-8,(2007),1101-1113. 10.1016/J.IJMACHTOOLS.2006.09.023
XVII. Tabata T., Masaki S. and Abe Y., “Analysis of Forging P/M Preforms” Journal of the Japan Society for Technology of Plasticity, 18, (1977), 373.
XVIII. Tabata T., Masaki S. and Hosokawa K., “A Compression Test to Determine the Coefficient of Friction in Forging P/M Preforms”, International Journal of Powder Metallurgy Powder Technology, 16, (1980), 149.

View | Download

Abstract:

This study investigates the influence of die velocity in sinter forging, particularly focusing on plastic deformation characteristics in the cold forging of axially symmetric components. Key factors such as material flow, inertia energy dissipation, and die load are examined. Results show that both energy dissipation and die load increase with higher die speeds. Utilizing an upper-bound approach with a simplified velocity field, theoretical results for the average die load are established. This analysis aims to improve the understanding of dynamic effects in sinter-forging cylindrical preforms, offering valuable insights for future research in this domain. Additionally, this paper provides a comprehensive review of the role of friction in metal forming processes, emphasizing the development of theoretical models to analyze tool-workpiece interfaces under varying friction conditions using the upper-bound method. The importance of understanding friction, particularly in forging processes, is highlighted. A composite friction mechanism in axisymmetric forging is introduced, accounting for the effect of platen speed. The significance of friction conditions on key factors such as applied load and plastic deformation is underscored, particularly considering the crucial interaction between adhesion and sliding.

Keywords:

Sinter forging,Upper bound,Die Load,Dynamical effect,sticking and sliding friction,

Refference:

I. Agrawal M., Jha A. K., Kumar S., “High-speed forging of hollow metal powder preforms”, Inst. Engrs. (I) J., 80 (1999),8.
II. Avitzur B., “Metal Forming Processes and Analysis”, McGraw Hill, New York, 1968.
III. Cost savings win the day for PM parts, Metal Powder Report, Vol. 56, Issue 7-8, (2001), 10-14.
IV. Jain Shrikant, Ranjan R. K. and Kumar Surender, “Fracturing and Deformation Characteristics of Aluminium Preform during Cold Forging at Low Strain Rates”. Int. J. of Scientific Engineering and Technology, 4(3) (2015),182-186. 10.17950/ijset/v4s3/314
V. Jha A. K and Kumar S., “Analysis of Axisymmetric Cold Processing of Metal Powder Preforms”, Journal of the Institution of Engineers (India), 65, (1985), 169.
VI. Jha A. K. and Kumar S., “Compatibility of sintered materials during cold forging”, International Journal of Materials and Product Technology, Vol. 9, Issue 4-6, (1994), 281-299. 10.1504/IJMPT.1994.036423
VII. Jha A. K., Kumar S., “Dynamic effects during a high-speed sinter-forging process”, International Journal of Machine Tools and Manufacture, Vol. 36, Issue 10, (1996), 1109-1122, ISSN 0890-6955.
10.1016/0890-6955(95)00122-0
VIII. Jha A. K., Kumar S., “Compatibility of sintered materials during cold forging”, International Journal of Materials and Product Technology’ 9, Issue 4-6, (2004), 281-299. 10.1504/IJMPT.1994.036423
IX. Jones P. K., “The technical and economic advantage of powder forged products”, Powder Metallurgy, Vol. 13, Issue 26, (1970), 114-129.
X. Kumar Parveen, Ranjan R. K., Kumar Rajive, “Mechanics of deformation during open die forging of sintered preform: Comparative study by equilibrium and upper bound methods”, ARPN Journal of Engineering and Applied Sciences, 6(6), (2011)83–93.
https://arpnjournals.com/jeas/research_papers/rp_2011/jeas_0611_515.pdf
XI. Kumar Parveen, Ranjan R. K., Kumar Rajive, “Investigations of an axisymmetric compound flow behavior of sintered preform: An upper bound approach”, International Journal of Pure and Applied Mathematics, Vol. 81, Issue 5, (2012), 671-691.
https://www.ijpam.eu/contents/2012-81-5/2/2.pdf
XII. Kumar Parveen, Ranjan R. K., Kumar Rajive, “Mathematical modelling of forging of sintered preform: Comparative study of open and closed die”, International Journal of Pure and Applied Mathematics, Vol. 82, Issue 2, (2012), 179-188. https://www.ijpam.eu/contents/2013-82-2/2/2.pdf
XIII. Kumar Parveen, Ranjan R. K., “Investigation on sintered preform with different geometrical shape”, AIP conference Proceedings, Volume 2142, 2019. 10.1063/1.5122620
XIV. Ranjan R. K. and Kumar S., “Effect of interfacial friction during forging of solid powder discs of large slenderness ratio”. Sadhana, 29, 535-543, 2004. 10.1007/BF02703260
XV. Rooks B. W., “The effect of die temperature on metal flow and die wear during high-speed hot forging”, In Proceedings of the Fifteenth International Machine Tool Design and Research Conference, Springer, (1975), 487-494
XVI. Singh S., Jha A. K., “Sintered preforms adds better value to aerospace components”, Journal of Aerospace Engineering, I. E. (I), 82, (2001), 1-6.
XVII. Singh Saranjit, Jha A.K., “Analysis of dynamic effects during high-speed forging of sintered preforms”, Journal of Materials Processing Technology, Volume 112, Issue 1, 2001,Pages 53-62, ISSN 0924-0136 10.1016/S0924-0136(00)00898-0
XVIII. Singh S., Jha A. K., “An energy analysis during forging of sintered truncated conical preform at high-speed”, Tamkang J. of Science and Engineering,7,(2004),227-236. http://jase.tku.edu.tw/articles/jase-200412-7-4-05
XIX. Singh Saranjit & Jha A. K. & Kumar Suhas. “Dynamic effects during sinter forging of axi-symmetric hollow disc preforms”, International Journal of Machine Tools and Manufacture, Vol. 47, Issue 7-8,(2007),1101-1113. 10.1016/J.IJMACHTOOLS.2006.09.023
XX. Tabata T., Masaki S. and Abe Y.,”Analysis of Forging P/M Preforms” Journal of the Japan Society for Technology of Plasticity, 18, (1977), 373.
XXI. Tabata T., Masaki S. and Hosokawa K., “A Compression Test to Determine the Coefficient of Friction in Forging P/M Preforms”, International Journal of Powder Metallurgy Powder Technology, 16, (1980), 149.

View | Download

Abstract:

This paper explores a ranking method for Type-2 Intuitionistic Fuzzy Numbers (T2IFNs). Initially, we discuss the concept of T2IFNs and their operational laws involving addition, multiplication, and exponentiation. Furthermore, we introduce prioritized average operators designed to solve multiple attribute group decision-making (MAGDM) problems under a T2IFN environment, considering varying priority levels for attributes and experts. Specifically, we examine the mathematical properties of the T2IFNs Prioritized Weighted Average (T2IFPWA) operators. Then, after we apply the closeness coefficient method to a normalized prioritized weighted averaging matrix to determine the final ranking of alternatives. To illustrate the feasibility and effectiveness of the proposed approach, a real-world application in the context of talent acquisition is presented. Finally, the alternatives are ranked according to their computed closeness coefficients.

Keywords:

MAGDM,Prioritized Weighted Average Operator,T2FS,T2IFS,

Refference:

I. Atanassov, K., and G. Gargov. “Interval-valued intuitionistic fuzzy sets.” *Fuzzy Sets and Systems*, vol. 31, no. 3, 1989, pp. 343–349. 10.1016/0165-0114(89)90205-4
II. Beliakov, Gleb, Ana Pradera, and Tomasa Calvo. *Aggregation Functions: A Guide for Practitioners*. Springer, 2007. 10.1007/978-3-540-73721-6
III. Bordogna, G., M. Fedrizzi, and G. Pasi. “A linguistic modeling of consensus in group decision making based on OWA operators.” *IEEE Transactions on Systems, Man, and Cybernetics – Part A: Systems and Humans*, vol. 27, no. 1, 1997, pp. 126–133. 10.1109/3468.553232
IV. Broumi, Said, and Florentin Smarandache. “Intuitionistic neutrosophic soft set.” *Journal of Information and Computing Science*, vol. 8, no. 2, 2013, pp. 130–140. https://digitalrepository.unm.edu/math_fsp/514
V. Fodor, J., J.-L. Marichal, and M. Roubens. “Characterization of the ordered weighted averaging operators.” *IEEE Transactions on Fuzzy Systems*, vol. 3, no. 2, 1995, pp. 236–240. 10.1109/91.388176
VI. Garg, Harish, and Kiran Kumar. “A novel approach for analyzing the efficiency of grey systems using a possibility degree-based method under interval-valued intuitionistic fuzzy environment.” *Complex & Intelligent Systems*, vol. 6, no. 3, 2020, pp. 487–504. https://link.springer.com/article/10.1007/s40747-020-00152-y
VII. Garg, Harish, and Manish Kumar. “Some prioritized aggregation operators based on Bonferroni mean under neutrosophic environment and their application in decision making.” *Computers & Industrial Engineering*, vol. 140, 2020, article no. 106279. https://www.sciencedirect.com/science/article/pii/S036083521930615X
VIII. Karnik, Nilesh N., and Jerry M. Mendel. “Operations on type-2 fuzzy sets.” *Fuzzy Sets and Systems*, vol. 122, no. 2, 2001, pp. 327–348. 10.1016/S0165-0114(00)00079-8
IX. Liu, Feilong, and Jerry M. Mendel. “Encoding words into interval type-2 fuzzy sets using the interval approach.” *IEEE Transactions on Fuzzy Systems*, vol. 19, no. 1, 2011, pp. 107–120. 10.1109/TFUZZ.2008.2005002
X. Liu, Xiang, and Hong Wang. “Multi-criteria decision making methods based on triangular interval type-2 fuzzy numbers.” *Mathematics*, vol. 7, no. 10, 2019, article no. 885. 10.3390/math7100885
XI. Mendel, Jerry M. “Type-2 fuzzy sets and systems: An overview.” *IEEE Computational Intelligence Magazine, vol. 2, no. 1, 2007, pp. 20–29. 10.1109/MCI.2007.380672
XII. Mendel, Jerry M., Robert I. John, and Feilong Liu. “Interval type-2 fuzzy logic systems made simple.” IEEE Transactions on Fuzzy Systems, vol. 14, no. 6, 2006, pp. 808–821. 10.1109/TFUZZ.2006.879986
XIII. Mitchell, H. B. “Pattern recognition using type-II fuzzy sets.” *Information Sciences*, vol. 170, no. 2, 2005, pp. 409–418. 10.1016/j.ins.2004.02.027
XIV. Türkşen, I. B. “Type-2 representation and reasoning for CWW.” *Fuzzy Sets and Systems*, vol. 127, no. 1, 2002, pp. 17–36. 10.1016/S0165-0114(01)00150-6
XV. Wang, X., and Z. Fan. “Fuzzy ordered weighted averaging operator and its applications.” *Fuzzy Systems and Mathematics*, vol. 17, no. 4, 2003, pp. 67–72.
XVI. Wu, Dongrui, and Jerry M. Mendel. “Aggregation using the linguistic weighted average and interval type-2 fuzzy sets.” *IEEE Transactions on Fuzzy Systems*, vol. 15, no. 6, 2007, pp. 1145–1161. 10.1109/TFUZZ.2007.896325
XVII. Wu, Dongrui, and Jerry M. Mendel. “Uncertainty measures for interval type-2 fuzzy sets.” *Information Sciences*, vol. 177, no. 23, 2007, pp. 5378–5393. 10.1016/j.ins.2007.07.012
XVIII. Xu, Zeshui. “Group decision making based on generalized fuzzy weighted aggregation operators.” *Applied Soft Computing*, vol. 8, no. 1, 2008, pp. 599–607. 10.1016/j.asoc.2007.05.008
XIX. Xu, Zeshui, and Hongchun Wang. “Distance and similarity measures for hesitant fuzzy sets.” *Information Sciences*, vol. 181, no. 11, 2011, pp. 2128–2138. 10.1016/j.ins.2011.01.028
XX. Xu, Zeshui, and Q. L. Da. “An overview of operators for aggregating information.” *International Journal of Intelligent Systems*, vol. 18, no. 9, 2003, pp. 953–969. 10.1002/int.10127
XXI. Xu, Zeshui, and Q. L. Da. “The ordered weighted geometric averaging operators.” *International Journal of Intelligent Systems*, vol. 17, no. 7, 2002, pp. 709–716. 10.1002/int.10045
XXII. Xu, Zeshui, and Ronald R. Yager. “Some geometric aggregation operators based on intuitionistic fuzzy sets.” *International Journal of General Systems*, vol. 35, no. 4, 2006, pp. 417–433. 10.1080/03081070600574353
XXIII. Xu, Zeshui, and Weiru Xia. “Distance and similarity measures for hesitant fuzzy linguistic term sets and their applications.” *Knowledge-Based Systems*, vol. 42, 2013, pp. 57–69. 10.1016/j.knosys.2013.08.002
XXIV. Yager, Ronald R. “On ordered weighted averaging aggregation operators in multi-criteria decision making.” *IEEE Transactions on Systems, Man and Cybernetics*, vol. 18, no. 1, 1988, pp. 183–190. 10.1109/21.87068
XXV. Yager, Ronald R. “Prioritized aggregation operators.” *International Journal of Approximate Reasoning*, vol. 48, no. 1, 2008, pp. 263–274. 10.1016/j.ijar.2007.08.009
XXVI. Ze-Shui, Xu. “A priority method for triangular fuzzy number complementary judgement matrix.” *Systems Engineering – Theory and Practice*, no. 10, 2003, pp. 1–13. 10.12011/1000-6788(2003)10-86
XXVII. Zadeh, L. A. “Fuzzy sets.” *Information and Control*, vol. 8, no. 3, 1965, pp. 338–353. 10.1016/S0019-9958(65)90241-X
XXVIII. Zadeh, L. A. “The concept of a linguistic variable and its application to approximate reasoning.” *Information Sciences*, vol. 8, no. 3, 1975, pp. 199–249. 10.1016/0020-0255(75)90036-5

View | Download

Abstract:

In this paper, we present an effective semi-analytical method for solving non-linear partial differential equations that arise in various scientific and engineering fields. The Homotopy Perturbation Method (HPM) combines the concepts of homotopy analysis and perturbation theory to obtain approximate solutions for diverse partial differential equations. Several numerical examples are presented to illustrate the accuracy and efficiency of the proposed method.

Keywords:

Homotopy Perturbation method,Partial differential equations,

Refference:

I. Abdel-Aty, A. H., M. Khater, R. A. Attia, and H. Eleuch. “Exact Traveling and Nano-Soliton Wave Solitons of the Ionic Waves Propagating along Microtubules in Living Cells.” Mathematics, vol. 8, no. 6, 2020, article 697.
II. Allahviranloo, Tofigh, Atefeh Armand, and Saeed Pirmohammadi. “Variational Homotopy Perturbation Method: An Efficient Scheme for Solving Partial Differential Equations in Fluid Mechanics.” Journal of Mathematics and Computer Science, vol. 9, no. 4, 2014, pp. 362–69.
III. Biazar, J., K. Hosseini, and P. Gholamin. “Homotopy Perturbation Method for Solving KdV and Sawada–Kotera Equations.” Journal of Applied Mathematics, vol. 6, 2009, pp. 23–29.
IV. Daga, A., and V. Pradhan. “A Novel Approach for Solving Burger’s Equation.” Applications & Applied Mathematics, vol. 9, 2014, pp. 541–52.
V. Foursov, M. V., and M. M. Maza. “On Computer-Assisted Classification of Coupled Integrable Equations.” Journal of Symbolic Computation, vol. 33, 2002, pp. 647–60.
VI. He, J. H. “Comparison of Homotopy Perturbation Method and Homotopy Analysis Method.” Applied Mathematics and Computation, vol. 156, 2004, pp. 527–39.
VII. Khater, M. M., B. Ghanbari, K. S. Nisar, and D. Kumar. “Novel Exact Solutions of the Fractional Bogoyavlensky–Konopelchenko Equation Involving the Atangana-Baleanu-Riemann Derivative.” Alexandria Engineering Journal, vol. 59, 2020, pp. 2957–67.
VIII. Liao, S. Beyond Perturbation: Introduction to Homotopy Analysis Method. CRC Press, 2003.
IX. Liang, S., and D. J. Jeffrey. “Comparison of Homotopy Analysis Method and Homotopy Perturbation Method Through an Evolution Equation.” Department of Mathematics, University of Western Ontario, 2009, pp. 1–12.
X. Maini, P. K., D. L. S. McElwain, and D. I. Leavesley. “Traveling Wave Model to Interpret a Wound Healing Cell Migration Assay for Human Peritoneal Mesothelial Cells.” Tissue Engineering, vol. 10, 2004, pp. 475–82.
XI. “Travelling Waves in a Wound Healing Assay.” Applied Mathematics Letters, vol. 17, 2004, pp. 575–80.
XII. Matinfar, M., M. Mahdavi, and Z. Raeisy. “The Implementation of Variational Homotopy Perturbation Method for Fisher’s Equation.” International Journal of Nonlinear Science, vol. 9, no. 2, 2010, pp. 188–94.
XIII. Park, C., M. M. Khater, A. H. Abdel-Aty, R. A. Attia, H. Rezazadeh, A. Zidan, and A. B. Mohamed. “Dynamical Analysis of the Nonlinear Complex Fractional Emerging Telecommunication Model with Higher–Order Dispersive Cubic–Quintic.” Alexandria Engineering Journal, vol. 59, 2020, pp. 1425–33.
XIV. Qin, H., M. Khater, and R. A. Attia. “Copious Closed Forms of Solutions for the Fractional Nonlinear Longitudinal Strain Wave Equation in Microstructured Solids.” Mathematical Problems in Engineering, 2020, article 3498796.
XV. Sherratt, J. A., and J. D. Murray. “Models of Epidermal Wound Healing.” Proceedings of the Royal Society B, vol. 241, 1990, pp. 29–36.
XVI. Simpson, M. J., K. K. Treloar, B. J. Binder, P. Haridas, K. J. Manton, D. I. Leavesley, D. L. S. McElwain, and R. E. Baker. “Quantifying the Roles of Cell Motility and Cell Proliferation in a Circular Barrier Assay.” Journal of the Royal Society Interface, vol. 10, 2013, article 20130007.
XVII. Xue, C., D. Lu, M. M. Khater, A. H. Abdel-Aty, W. Alharbi, and R. A. Attia. “On Explicit Wave Solutions of the Fractional Nonlinear DSW System via the Modified Khater Method.” Fractals, vol. 28, no. 2, 2020, article 2040034.
XVIII. Yue, C., D. Lu, M. M. Khater, A. H. Abdel-Aty, W. Alharbi, and R. A. Attia. “On Explicit Wave Solutions of the Fractional Nonlinear DSW System via the Modified Khater Method.” Fractals, vol. 28, no. 2, 2020, article 2040034.

View | Download

Abstract:

When it comes to dealing with nonlinear equations, numerical methods play a crucial role. Still, many of these methods come with limitations such as guaranteeing actual convergence, high computational costs, or strong dependence on derivatives. Traditional techniques, in particular, tend to struggle when the first derivative is close to zero or when they require second or third derivatives, which adds layers of complexity. The study presents a new iterative approach to overcome these challenges. It achieves a reliable second-order convergence and can handle both real and complex rootseven in situations where the first derivative approaches zero. The method starts with an initial guess, w0 ∈ C, and improves it step-by-step, gradually zeroing in on a solution. Its flexibility allows it to be applied to a broad range of equations. One of the key advantages is that it doesn’t depend on higher-order derivatives, which helps in maintaining a balance between computational efficiency and accuracy.. Interestingly, the method also manages to find complex roots even when the initial guess is entirely real, something many other methods struggle with. To evaluate how well the method works, experiments were conducted using Python version 3.10.12. The results shown in tables and graphs illustrate how the method converges over a set number of steps. Overall, this technique offers a reliable and practical alternative to conventional numerical methods, particularly for tackling nonlinear problems involving complex solutions

Keywords:

Nonlinear,Complex root,Iterative numerical methods,Second-order convergence,innovation,

Refference:

I. Abbasbandy, Saeid. “Extended Newton’s Method for a System of Nonlinear Equations by Modified Adomian Decomposition Method.” Applied Mathematics and Computation, vol. 170, no. 1, 2005, pp. 648–656. https://doi.org/10.1016/j.amc.2004.11.061.
II. Abdullah, S., Choubey, N., and Dara, S. “Dynamical Analysis of Optimal Iterative Methods for Solving Nonlinear Equations with Applications.” Journal of Applied Analysis and Computation, vol. 14, no. 6, 2024, pp. 3349–3376.
III. Al-Jawary, M. A., et al. “Three Iterative Methods for Solving Second Order Nonlinear ODEs Arising in Physics.” Journal of King Saud University-Science, vol. 32, no. 1, 2020, pp. 312–323.
IV. Cordero, A., Soleymani, F., and Torregrosa, J. R. “Dynamical Analysis of Iterative Methods for Nonlinear Systems.” Applied Mathematics and Computation, vol. 244, 2014, pp. 398–412.
V. Dehghan, M., and Shirilord, A. “Three-Step Iterative Methods for Numerical Solution of Systems of Nonlinear Equations.” Engineering Computations, vol. 38, 2022, pp. 1015–1028.
VI. Hansen, E., and Patrick, M. “A Family of Root Finding Methods.” NumerischeMathematik, vol. 27, 1977, pp. 257–269. 10.1007/BFb0062109.
VII. Inderjeet, and Bhardwaj, R. “A New Iterative Newton-Raphson Technique for the Numerical Simulation of Nonlinear Equations.” Journal of Integrated Science and Technology, vol. 13, no. 4, 2025, pp. 1080.
VIII. Ivanov, I. G., and Yang, H. “On the Iterative Methods for the Solution of Three Types of Nonlinear Matrix Equations.” Mathematics, vol. 11, no. 21, 2023, pp. 4436.
IX. Ji-Huan, He. “A Review on Some New Recently Developed Nonlinear Analytical Techniques.” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 1, 2000, pp. 51–70.
X. Kaur, H., and Malhotra, R. “Comparative Analysis of a Redundant System Subject to Inspection of a Manufacturing Plant.” Journal of Mechanics of Continua and Mathematical Sciences, 2024, pp. 85–105.
XI. Kaur, H., and Malhotra, R. “Profit Analysis of a System of Non-Identical Units with Varying Demand.” Palestine Journal of Mathematics, vol. 14, Special Issue I, 2025, pp. 183–197.
XII. Kaur, H., and Malhotra, R. “Stochastic Analysis of a Two-Unit Standby Autoclave System with Inspection and Varying Demand.” Journal of Mechanics of Continua and Mathematical Sciences, 2024, pp. 147–163.
XIII. King, R. F. “A Family of Fourth-Order Methods for Nonlinear Equations.” SIAM Journal on Numerical Analysis, vol. 10, no. 5, 1973, pp. 876–879.
XIV. Maheshwari, A. K. “A Fourth Order Iterative Method for Solving Nonlinear Equations.” Applied Mathematics and Computation, vol. 211, no. 2, 2009, pp. 383–391.
XV. Malhotra, R., and Taneja, G. “Comparative Analysis of Two Stochastic Models Subjected to Inspection and Scheduled Maintenance.” International Journal of Software Engineering and Its Applications, vol. 9, no. 10, 2015, pp. 179–188.
XVI. Malhotra, R., and Taneja, G. “Comparative Analysis of Two Stochastic Models with Varying Demand.” International Journal of Applied Engineering Research, vol. 10, no. 17, 2015, pp. 37453–37460.
XVII. Melman, A. “The Double-Step Newton Method for Polynomials with All Real Zeros.” Applied Mathematics Letters, vol. 20, no. 6, 2007, pp. 671–675.
XVIII. Neta, B. “A Sixth-Order Family of Methods for Nonlinear Equations.” International Journal of Computer Mathematics, vol. 7, no. 2, 1979, pp. 157–161.
XIX. Noor, M. A., et al. “On Iterative Methods for Nonlinear Equations.” Applied Mathematics and Computation, vol. 183, 2006, pp. 128–133.
XX. Ostrowski, A. M. Solution of Equations and Systems of Equations. Academic Press, 1966.
XXI. Paniconi, C., et al. “Numerical Evaluation of Iterative and Noniterative Methods for the Solution of the Nonlinear Richards Equation.” Water Resources Research, vol. 27, no. 6, 1991, pp. 1147–1163.
XXII. Parhi, S. K., and Gupta, D. K. “A Sixth-Order Method for Nonlinear Equations.” Applied Mathematics and Computation, vol. 203, no. 1, 2008, pp. 50–55.
XXIII. Ramos, J. I. “On the Variational Iteration Method and Other Iterative Techniques for Nonlinear Differential Equations.” Applied Mathematics and Computation, vol. 199, 2008, pp. 39–69.
XXIV. Sharma, J. R., and Guha, R. K. “A Family of Modified Ostrowski Methods with Accelerated Sixth-Order Convergence.” Applied Mathematics and Computation, vol. 190, no. 1, 2007, pp. 111–115.
XXV. Siwach, A., and Malhotra, R. “Complex Roots-Finding Method of Nonlinear Equations.” Contemporary Mathematics, vol. 5, no. 3, 2024, pp. 2848–2857.
XXVI. Traub, J. F. Iterative Methods for the Solution of Equations. Prentice Hall, 1964.
XXVII. Wang, X., et al. “A Family of Newton Type Iterative Methods for Solving Nonlinear Equations.” Algorithms, vol. 8, no. 3, 2015, pp. 786–798.
XXVIII. Weerakoon, S., and Fernando, T. G. I. “A Variant of Newton’s Method with Accelerated Third-Order Convergence.” Applied Mathematics Letters, vol. 13, no. 8, 2000, pp. 87–93.

View | Download

Abstract:

In this study, researchers propose an innovative numerical approach to solve non-linear equations for real as well as complex roots. The approach, initiated with an initial guess in the complex plane, iteratively converges towards solutions. A notable feature is its ability to accurately identify complex roots even when initialized with a real number. The method demonstrates second-order convergence, with its efficacy evaluated through quantifying the number of iterations needed for convergence. Using Python 3.10.9, experiments were conducted to evaluate its effectiveness across various numerical problems. Results were presented in tabular format, supplemented by graphical representations. Furthermore, the study examines the method's computational efficiency by analyzing CPU time and introducing an efficiency index.

Keywords:

High-order transcendental equations,Nonlinear,Complex root-finding algorithms,Convergence,Innovation,

Refference:

I. Abbasbandy, Saeid. “Extended Newton’s Method for a System of Nonlinear Equations by Modified Adomian Decomposition Method.” Applied Mathematics and Computation, vol. 170, no. 1, 2005, pp. 648–656. 10.1016/j.amc.2004.11.061.
II. Ahmad, N., and Singh, V. P. “A New Efficient Four-Step Iterative Method with 36th-Order Convergence for Solving Nonlinear Equations.” Journal of Nonlinear Modeling and Analysis, vol. 5, no. 2, 2023, pp. 1-8.
III. Al-Obaidi, R. H., and M. T. Darvishi. “Constructing a Class of Frozen Jacobian Multi-Step Iterative Solvers for Systems of Nonlinear Equations.” Mathematics, vol. 10, no. 16, 2022, pp. 2952–2965. 10.3390/math10162952.
IV. Bayrak, M. A., A. Demir, and E. Ozbilge. “On Fractional Newton-Type Method for Nonlinear Problems.” Journal of Mathematics, 2022. 10.1155/2022/3768720.
V. Fang, X., Q. Ni, and M. Zeng. “A Modified Quasi-Newton Method for Nonlinear Equations.” Journal of Computational and Applied Mathematics, vol. 328, 2018, pp. 44–58. 10.1016/j.cam.2017.07.012.
VI. Gong, W., Z. Liao, X. Mi, L. Wang, and Y. Guo. “Nonlinear Equations Solving with Intelligent Optimization Algorithms: A Survey.” Complex System Modeling and Simulation, vol. 1, no. 1, 2021, pp. 15–32.
VII. Hansen, E., and M. Patrick. A Family of Root Finding Methods. Springer-Verlag, 1977, pp. 257–269. 10.1007/BFb0062109.
VIII. Kaur, H., and Malhotra, R. “Comparative Analysis of a Redundant System Subject to Inspection of a Manufacturing Plant.” Journal of Mechanics of Continua and Mathematical Sciences, 2024, pp. 85–105. 10.26782/jmcms.spl.11/2024.05.00007
IX. Kaur, H., and Malhotra, R. “Profit Analysis of a System of Non-Identical Units with Varying Demand.” Palestine Journal of Mathematics, vol. 14, Special Issue I, 2025, pp. 183–197.
X. Kaur, H., and Malhotra, R. “Stochastic Analysis of a Two-Unit Standby Autoclave System with Inspection and Varying Demand.” Journal of Mechanics of Continua and Mathematical Sciences, 2024, pp. 147–163. 10.26782/jmcms.spl.11/2024.05.00011
XI. Maheshwari, A. K. “A Fourth-Order Iterative Method for Solving Nonlinear Equations.” Applied Mathematics and Computation, vol. 211, no. 2, 2009, pp. 383–391. 10.1016/j.amc.2009.02.048.
XII. Malhotra, R. and Taneja, G. “Comparative Analysis of Two Stochastic Models Subjected to Inspection and Scheduled Maintenance.” International Journal of Software Engineering and Its Applications, vol. 9, no. 10, 2015, pp. 179–188.
XIII. Malhotra, R. and Taneja, G. “Comparative Analysis of Two Stochastic Models with Varying Demand.” International Journal of Applied Engineering Research, vol. 10, no. 17, 2015, pp. 37453–37460.
XIV. Melman, A. “The Double-Step Newton Method for Polynomials with All Real Zeros.” Applied Mathematics Letters, vol. 20, no. 6, 2007, pp. 671–675. 10.1016/j.aml.2006.07.007.
XV. Mitlif, R. J. “New Iterative Method for Solving Nonlinear Equations.” Baghdad Science Journal, vol. 11, no. 4, 2014, pp. 1649–1654.
XVI. Noor, M. A., and M. Waseem. “Some Iterative Methods for Solving a System of Nonlinear Equations.” Computers and Mathematics with Applications, vol. 57, no. 1, 2009, pp. 101–106. 10.1016/j.camwa.2008.08.007.
XVII. Ostrowski, A. M. Solution of Equations and Systems of Equations. Academic Press, 1966.
XVIII. Parhi, S. K., and D. K. Gupta. “A Sixth-Order Method for Nonlinear Equations.” Applied Mathematics and Computation, vol. 203, no. 1, 2008, pp. 50–55. 10.1016/j.amc.2008.04.033.
XIX. Popovski, D. B. “A Note on Neta’s Family of Sixth-Order Methods for Solving Equations.” International Journal of Computer Mathematics, vol. 10, no. 1, 1981, pp. 91–93. 10.1080/00207168108803241.
XX. Roman, C. “A Family of Newton-Chebyshev-Type Methods to Find Simple Roots of Nonlinear Equations and Their Dynamics.” Communications in Numerical Analysis, vol. 2017, no. 2, 2017, pp. 172–185.
XXI. Sharma, H., R. Behl, M. Kansal, and H. Ramos. “A Robust Iterative Family for Multiple Roots of Nonlinear Equations: Enhancing Accuracy and Handling Critical Points.” Journal of Computational and Applied Mathematics, vol. 444, 2024, p. 115795. 10.1016/j.cam.2023.115795.
XXII. Sharma, J. R., and R. K. Guha. “A Family of Modified Ostrowski Methods with Accelerated Sixth-Order Convergence.” Applied Mathematics and Computation, vol. 190, no. 1, 2007, pp. 111–115. 10.1016/j.amc.2007.01.083.
XXIII. Sharma, J. R., R. K. Guha, and R. Sharma. “Some Variants of Hansen-Patrick Method with Third and Fourth Order Convergence.” Applied Mathematics and Computation, vol. 214, no. 1, 2009, pp. 171–177. 10.1016/j.amc.2009.03.063.
XXIV. Sharma, J. R., and R. Sharma. “Modified Jarratt Method for Computing Multiple Roots.” Applied Mathematics and Computation, vol. 217, no. 2, 2010, pp. 878–881. 10.1016/j.amc.2010.06.021.
XXV. Sharma, R., and A. Bahl. “An Optimal Fourth-Order Iterative Method for Solving Nonlinear Equations and Their Dynamics.” Journal of Complex Analysis, 2015, Article ID 935723. 10.1155/2015/935723.
XXVI. Siwach, A., and R. Malhotra. “Complex Roots-Finding Method of Nonlinear Equations.” Contemporary Mathematics, vol. 5, no. 3, 2024, pp. 2848–2857.
XXVII. Singh, S., and D. K. Gupta. “A New Sixth-Order Method for Nonlinear Equations in R.” The Scientific World Journal, vol. 2014, Article ID 509579, 2014. 10.1155/2014/509579.
XXVIII. Traub, J. F. Iterative Methods for the Solution of Equations. Prentice Hall, 1964.

View | Download