Journal Vol – 18 No -7, July 2023



Jawad Kadhim Tahir



In this article, a numerical solution to the growth-diffusion problem is investigated by obtaining the results of computational experiments for the non-homogeneous growth-diffusion problem and finding its approximate solution by using the modified finite difference method. In this article, a numerical study is carried out by the modified finite difference method. The numerical scheme used a second-order central difference in space with a first-order in time.


growth-diffusion problem,modified finite difference method,central difference,non-classical variational,


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Poulomi Mullick, Goutam Sengupta, Sujoy Biswas, Nabonita Pal, Prosenjit Pramanick, Sana Ahmed, Sufia Zaman, Abhijit Mitra



The Indian Sundarbans are noted for luxuriant mangrove diversity that is known to scrub carbon dioxide from the atmosphere. Precise estimation of the biomass of these species is necessary for evaluating the carbon storage pattern in the mangroves of the lower Gangetic belt. The plant biomass estimation was carried out for an average of 25 trees in 15 (10 m × 10 m) plots from the intertidal mudflats of Chemaguri (southeast portion of Sagar Island) in low tide conditions from 10th to 15th September 2022. The estimated biomass was of the order Sonneratia apetala > Avicennia alba > Avicennia marina > Excoecaria agallocha > Avicennia officinalis. The stem, branch, and leaf biomass of each species were converted into carbon by multiplying with a factor of 0.45 as per the standard procedure. The deviations observed in the results obtained from both studies call for the standardization of the process.


Carbon storage,mangroves,Above Ground Biomass (AGB),Below Ground Biomass (BGB),


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M. Al-Amin , M. Nurul Islam



In this study, the nonlinear Landau-Ginsberg-Higgs (LGH) model is proposed and examined. The stated model is applied to analyze superconductivity and drift cyclotron waves in radially inhomogeneous plasma for coherent ion-cyclotron waves. This is undeniably a robust mathematical model in real-world applications. The generalized exponential rational function method (GERFM) is utilized to extract the suitable, useful, and further general solitary wave solutions of the LGH model via the traveling wave transformation. Furthermore, we investigate the effects of wave velocity in a particular time limit through a graphical representation of the examined solutions of the model to understand the dynamic behavior of the system. The attained results confirm the effectiveness and reliability of the considered scheme


The nonlinear Landau-Ginsberg-Higgs (LGH) model,the generalized exponential rational function method (GERFM),the traveling wave transformation,the soliton solutions,


I. A.S.H.F. Mohammed, H.O. Bakodah, M.A. Banaja, A.A. Alshaery, Q. Zhou, A. Biswas, P. Seithuti. Moshokoa, M.R. Belic, Bright optical solitons of Chen-Lee-Liu equation with improved Adomian decomposition method, Optik, 181:964-970, 2019.

II. A. Zafar, H. Rezazadeh, K.K. Ali, On finite series solutions of conformable time-fractional Cahn-Allen equation, Nonlin. Eng., 9(1):194-200, 2020.

III. A. Yusuf, M. Inc, A.I. Aliyu, D. Baleanu, Optical Solitons Possessing Beta Derivative of the Chen-Lee-Liu Equation in Optical Fibers, Fron. Phys., 7, 34, 2019.

IV. A.S.H.F. Mohammed, H.O. Bakodah, Approximate Solutions for Dark and Singular Optical Solitons of Chen-Lee-Liu Model by Adomian-based Methods, Int. J. Appl. Comput. Math., 7, 98, 2021.
V. A. Bekir, O. Unsal, Exact solutions for a class of nonlinear wave equations by using first integral method, Int. J. Nonlin. Sci., 15(2):99-110, 2013.

VI. A. Irshad, S.T. Mohyud-Din, N. Ahmed, U. Khan, A new modification in simple equation method and its applications on nonlinear equations of physical nature, Results Phys., 7:4232-40, 2017.

VII. A.C. Cevikel, E. Aksoy, O. Guner, A. Bekir, Dark bright soliton solutions for some evolution equations, Int. J. Nonlin. Sci., 16(3):195-202, 2013.

VIII. A. Iftikhar, A. Ghafoor, T. Zubair, S. Firdous, S.T. Mohyud-Din, (????′????⁄,1????⁄)-Expansion method for traveling wave solutions of (2+1) dimensional generalized KdV, Sine Gordon and Landau-Ginzburg-Higgs equations, Sci. Res. Essays., 8(28):1349-59, 2013.

IX. B. Ghanbari, D. Baleanu, M.A. Qurashi, New Exact Solutions of the Generalized Benjamin-Bona-Mahony Equation, Symmetry, 11(1):20, 2019.

X. B. Ghanbari, M.S. Osman, D. Baleanu, Generalized exponential rational function method for extended Zakharov Kuzetsov equation with conformable derivative, Mod. Phy. Lett. A., 34,1950155,16pp, 2019.

XI. B. Ghanbari, M. Inc, A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear schrödinger equation, Eur. Phys. J. Plus, 133(4):142, 18pp, 2018.

XII. E.H.M. Zahran, M.M.A. Khater, Modified extended tanh-function method and its applications to the Bogoyavlenskii equation, Appl. Math. Model, 40(3):1769-1775, 2017.

XIII. F.S. Khodadad, S.M.M. Alizamini, B. Günay, L. Akinyemi, H. Rezazadeh, I. Mustafa, Abundant optical solitons to the Sasa-Satsuma higher-order nonlinear Schrödinger equation, Opt. Quant. Elec., 53, 702, 2021. XIV. H. Rezazadeh, A. Korkmaz, M. Eslami, S.M.M. Alizamini, A large family of optical solutions to Kundu-Eckhaus model by a new auxiliary equation method, Opt. Quant. Elec., 51(84), 2019.

XV. H.M. Baskonus, H. Bulut, T.A. Sulaiman, New complex hyperbolic structures to the Lonngren-Wave equation by using Sine-Gordon expansion method, App. Math. Non-lin. Sci., 4(1):129-138, 2019.

XVI. H.M. Baskonus, H. Bulut, A. Atangana, On the complex and hyperbolic structures of the longitudinal wave equation in a magneto-electro-elastic circular rod, Smart Mater. Struct., 25(3):035022, 2016.

XVII. K. Ahmad, K. Bibi, M.S. Arif, K. Abodayeh, New Exact Solutions of Landau-Ginzburg-Higgs Equation Using Power Index Method, J. Func. Spa., 4351698, 6pp, 2023.

XVIII. M.M. El-Borai, W.G. El-Sayed, R.M. Al-Masroub, Exact solutions for time fractional coupled Whitham-Broer-Kaup equations via exp-function method, Int. Res. J. Eng. Tech., 2(6):307-315, 2015.

XIX. M. Al-Amin, M.N. Islam, O.A. Ilhan, M.A. Akbar, D. Soybas, Solitary Wave Solutions to the Modified Zakharov-Kuznetsov and the (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Models in Mathematical Physics, J. Math., 2022, 5224289, 16pp, 2022.

XX. M. Al-Amin, M.N. Islam, M.A. Akbar, Adequate wide-ranging closed-form wave solutions to a nonlinear biological model, Par. Diff. Equ. App. Math., 2021(4):100042, 2021.

XXI. M. Al-Amin, M.N. Islam, M.A. Akbar, The closed-form soliton solutions of the time-fraction Phi-four and (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff model using the recent approach, Par. Diff. Equ. App. Math., 2022(5):100374, 2022.

XXII. M.N. Islam, M.A. Akbar, Closed form solutions to the coupled space-time fractional evolution equations in mathematical physics through analytical method, J. Mech. Cont. Math. Sci., 13(2):1-23, 2018.

XXIII. M.A. Akbar, N.H.M. Ali, E.M.E. Zayed, Abundant exact traveling wave solutions of generalized Bretherton equation via improved (????′????⁄)-expansion method, Com. Theo. Phys., 57(2012):173-178, 2012.

XXIV. M.A.E. Abdel Rahman, H.A. Alkhidhr, Closed-form solutions to the conformable space-time fractional simplified MCH equation and time fractional Phi-4 equation, Results Phys., 18, 103294, 2020.

XXV. M. Bilal, W. Hu, J. Ren, Different wave structures to the Chen-Lee-Liu equation of monomode fibers and its modulation instability analysis, Eur. Phys. J. Plus, 136(4):385, 2021.

XXVI. M.E. Islam, M.A. Akbar, Stable wave solutions to the Landau-Ginzburg-Higgs equation and the modified equal width wave equation using the IBSEF method, Arab J. Basic Appl. Sci., 27(1):270-278, 2020.

XXVII. M.R. Ali, M.A. Khattab, S.M. Mabrouk, Travelling wave solution for the Landau-Ginburg-Higgs model via the inverse scattering transformation method, Nonlin. Dyn., 111:7687-7697, 2023.

XXVIII. M.N. Islam, O.A. İlhan, M.A. Akbar, F.B. Benli, D. Soybaş, Wave propagation behavior in nonlinear media and resonant nonlinear interactions, Com. Nonlin. Sci. Num. Simul., 108, 106242, 2022.

XXIX. N. Ozdemir, H. Esen, A. Secer, M. Bayram, A. Yusuf, T.A. Sulaiman, Optical Soliton Solutions to Chen Lee Liu model by the modified extended tanh expansion scheme, Optik, 245, 167643, 2021.

XXX. O.A. Ilhan, M.N. Islam, M.A. Akbar, Construction of functional closed form wave solutions to the ZKBBM equation and the Schrodinger equation, Iranian J. Sci. Tech. Transac. Mech. Eng., 2020, 14pp, 2020.

XXXI. O.G. Gaxiola, A. Biswas, W-shaped optical solitons of Chen-Lee-Liu equation by Laplace-Adomian decomposition method, Opt. Quan. Electr., 50, 314:1-11, 2018.

XXXII. R. Roy, M.A. Akbar, A.R. Seadawy, D. Baleanu, Search for adequate closed form wave solutions to space-time fractional nonlinear equations, Par. Diff. Equ. App. Math., 2021(4):100025, 2021.

XXXIII. S.J. Chen, X. Lü, X.F. Tang, Novel evolutionary behaviors of the mixed solutions to a generalized Burgers equation with variable coefficients, Commun. Nonlin. Sci. Num. Simul., 95, 105628, 2021.

XXXIV. S. Albosaily, W.W. Mohammed, A.E. Hamza, M. El-Morshedy, H. Ahmad, The exact solutions of the stochastic fractional space Allen-Cahn equation, Open Phy., 20(1):23-29, 2022. XXXV. W.X. Ma, L. Zhang, Lump solutions with higher-order rational dispersion relations, Pram. J. Phys., 94(43), 2020.

XXXVI. W.X. Ma, Y. Zhang, Y. Tang, Symbolic computation of lump solutions to a combined equation involving three types of nonlinear terms, East Asian J. Appl. Math., 10(4):732-745, 2020.

XXXVII. W. Gao, H. Rezazadeh, Z. Pinar, H.M. Baskonus, S. Sarwar, G. Yel, Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique, Opt. Qua. Elec., 52(1), 2020.

XXXVIII. W.P. Hu, Z.C. Deng, S.M. Han, W. Fa, Multi symplectic Runge-Kutta method for Landau-Ginzburg-Higgs equation, Appl. Math. Mech., 30(8):1027-34, 2009.

XXXIX. Y. Yildirim, Optical solitons to Chen-Lee-Liu model in birefringent fibers with trial equation approach, Optik, 183:881-886, 2019.

XL. Y. Liu, J. Roberts, Y. Yan, A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes, Int. J. Com., 95(6-7):1151-1169, 2017.

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Abdul Qadir Mugheri, Asif Ali Shaikh, Shafqat Shahzoor Chandio Baloch



This research paper presents a novel and efficient explicit numerical technique for modeling advection diffusion reactions in an opened uniform flow from a one-dimensional perspective. The proposed hybrid scheme combines the benefits of explicit finite difference schemes, resulting in an accurate and fast solution for the advection-diffusion equation in water stream problems. The effectiveness of the scheme is demonstrated through its successful implementation in the solution of the water quality problems, where the advection-diffusion equation plays a crucial role. The results obtained using this technique show improved accuracy and computational efficiency. Overall, this research offers a valuable contribution to the field of numerical modeling in water quality and provides a useful tool for researchers and practitioners working in the area of approximating the one-dimensional diffusion equation for the measurement of pollutant concentration.


Explicit,Finite Difference,One Dimensional Advection Diffusion Equation,Uniform Flow,


I. Febi Sanjaya and Sudi Mungkasi, : “A simple but accurate explicit finite difference method for the advection-diffusion equation” International Conference on Science and Applied Science 2017 IOP Publishing, IOP Conf. Series: Journal of Physics: Conf. Series 909 (2017) 012038. 10.1088/1742-6596/909/1/012038.
II. Halil Karahan, : “Solution of Weighted Finite Difference Techniques with the Advection_Diffusion Equation Using Spreadsheets”. 2008 Wiley Periodicals, Inc. Comput Appl Eng Educ 16: 147_156, 2008; Published online in Wiley Inter Science ( 10.1002/cae.20140.
III. Inasse EL Arabi, Anas Chafi and Salaheddine Kammouri Alami, : “ Numerical simulation of the advection-diffusion-reaction equation using finite difference and operator splitting methods Application on the 1D transport problem of contaminant in saturated porous media”. E3S Web of Conferences 351, 01003 (2022). 10.1051/e3sconf/202235101003ICIES’22
IV. Mehdi Dehghan, : “Weighted Finite Difference Techniques For The One-dimensional Advection-Diffusion Equation”, ELSEVIER Applied Mathematics and computation 147 (2004) 307-319.
V. Pawarisa, Samalerk and Nopparat Pochai, : “A Saulyev Explicit Scheme for an One-Dimensional Advection-Diffusion-Reaction Equation in an Opened Uniform Flow Stream”. Thai Journal of Mathematics Volume 18 Number 2 (2020) Pages 677-683.
VI. Piyada, Phosri, NopparatPochai, : “Explicit Finite Difference Techniques for a One-Dimensional Water Pollutant Dispersion Model in a Stream”. PIYADA PHOSRI et al. 10.5013/IJSSST.a.21.03.01 1.1 ISSN: 1473-804x online, 1473-8031 print.
VII. Samalerk, Pawarisa and NopparatPochai, : “Numerical Simulation Of A One-Dimensional water Quality Model In A Stream Using A Saulyev Technique With Quadatic Interpolated Initial Boundary Conditions”. Hindawi Abstract and Applied Analysis Volume 2018, Article ID 1926519, 7 pages. 10.1155/2018/1926519.

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