Journal Vol – 4 No -1, July 2009



A.De , M. Chaudhuri



I. The abstract of this paper is Nowinski obtained the thermo-elostic stresses and displacements in spherical shells and solid spheres with temperature deyendent propreties. Following Gibson. The present author has obtained the thermo-elostic stresses in an in-homogeneous spherical shell where the poisson's ratio, the co-efficieent of exponential thermal expansion very exponentially with the radial distance r from the centre of the shell.


thermal stress,spherical shells, thermo – electric stresses,


I. Erdelyi, M.O.T. (1955) “Higher Trancendental Function”, McGraw-Hill Book Company, pp.40.

II. Gibson, R.E. (1969). “The thermo-elastic stresses in an in-homogeneous spherical shell” , ZAMP vol.20, pp.619.

III. Love, A.E.H. (1952) “A treatise on the mathematical theory of elasticity”, Cambridge UniversityPress, 4th Edn.

IV. MOllah, S.A. (1990) “Tharmal stresses in a non-homogeneous thin rotating circular disk having transient shearing stress applied on the outer edge” Ganit Journal of Bangladesh Mathematical Society Vol.10, pp.59-65.

V. Nowinski, J. (1959) “The thermo-elastic stresses and displacements in spherical shells and solid spheres” ZAMP, vol.10,pp.565.

VI. Saherical, B.K. (1965) “A stress distribution in a thin rotaiing circular dise having transient shearing stress applied on the outer edge”, Jour. Frank. Inst.vol.281, pp.315.

VII. Timoshenko, S. and Goodier, J.N. (1966)” Theory of elasticity” 4th Edn. McGraw Hill Book Co. New York, pp.406-434.

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Utpal Kumar Mandal



Large amplitude (nonlinar) free vibration analysis of thin shallow spherical elastic shalls of vairable thickness with tangentially clamped immovable edges has boon performed by using both (i) coupled governing differential equations derived in the Von Karman sense in trimes of displacement components as well as (ii) decoupled nonlinear governing differential equations on the basis of Berger approximation (i.e. neglection second strain invariant e2) derived from energy expression applying Hamilton`s principal and Euler`s variational equations. The governing differential equations are solved by Galerkin error minimizing technique incorporating clamped immovable edge conditions. A parametric study is presented to understand the effects of various parameters on nonlincear dynamic behavior of such structures and the same reveals some interesting features.


non linear vibration,spherical elastic shell,Berger approximation,Galerkin error ,


I. R. Archer and S. Lang, “Nonlinear dynamic behavior of shallow spherical shells, ” AIAAJ., no.2, pp.30-36, 1969.

II. J. Ramachandran, “Vibration of shallow spherical shells at large amplitudes” Journal of Applied Mechanics, ASME, vol.41, no.3, pp.811-812, 1974.

III. J. Ramachandran, “Large amplitiude vibration of shallow spherical shell with concentrated mass,” Journal of Applied Mechanics, ASME, vol.43, no.2, pp. 363-365, 1976.

IV.  P. Biswas, “Nonlinear Vibrations of a shallow shell of Variable Thickness,” Transactions of 11th International Conference on structural Mechanics in Reactor Technology (SMIRT-11), Tokyo, Japan, volSD2,05/5, pp.491-494. 1991.

V.U.K. Mandal and P.Biswas, “Nonliner tharmal vibrations on elastic shallow spherical shall under liner and parabolic temperature distributions,” Journal of Applied Mechanics, ASME, vol.66, np.3, pp.814-815.1991.

VI. Ghassan Odeh, “Nonliner dynamics of shallow Spherical caps subjected to peripheral, Netherlands, vol.33,pp.155-164, 2003.

VII. Wang Tono-gang and Dai Shi-liang, “Thermoelastically coupled axisymmetric nonlinear vibration of shallow spherical and conical shells,” Applied Mathematics and Mechanics, vol.24, no.4, pp.430-439, 2004.

VIII. W.A. Nash and J.R. Modeer, “Certain Approximate Analysis of the Nonlinear Behavior of Plates and Shallow Shells,” proceedings of Symposium on the Theory oh Thin Elastic Shells, Delft, The Netherlands, pp.331-353, 1969.

IX. A.P. Bhattacharyee, “Nonlinear Flexural Vibrations of Thin Shallow translational Shell, ” Journal of mechanics, ASME, vol.43, no.1, pp.180-181, 1976.

X. G.C. Sinharay And B.Banerjee, ” A new Approach to Large Deflection Analysis of Spherical and Cylindrical Shells under Thermal load,” Mechanics research Communication, vol.12, no.2, pp.53-64. 1985.

XI. G.C. Sinharay And B.Banerjee, ” Large Amplitude Free Vibrations of shallow spherical shell and Cylindrical Shell- A New Approach,” International Journal of Nonlinear Mechanics, vol.20, no.2, pp. 69-78., 1985.

XII. H.m. Berher, ” A new approachto the analisis of large Deflections of Plates,” Journal of Applied Mechanics, ASME, vol.22, pp.563-572, 1955.

XIII. B. Budiansky, “Buckling of Clamped Shallow spherical shell,” Proceedings of Symposium on the Theory of Thin Elastic Shalls, Delft, The Netherlands, pp.64-94, 1959.

XIV. S.P. Timoshenko and S. Woinowskty-Krieger, ” Theory of Plates and Shells, McGraw Hill, New York, 1959.

XV. U.K. Mandal, ” Nonlinear Vibrations of structures including Thermal Loading, ” Ph.D.Thesis, University of North Bengal, West Bengal India,2006.


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Goutam Chakraborty, Supriya Panja



In this paper oscillatory Couette Flow of a visco-elastic Rivlin-Ericknes fluid thoruge a porous medium within two non-conducting paralled plates in presence of a transverse unifrom magnatic field in a rotating system has been studied.


hydromagnetic Couette flow, visco-elstic fluid,Rivlin –Ericksen fluid,


I. Couette, M. (1890)- Ann.Chin, Phys., 21,433-510.

II. Tikekar, V.G. (1968) – Jour. Ind. Inst. Sci., 50,244.

III. Sengupta, P.R. and Ray, T.K. (1990) – Arch. Mech., 42,6, 717.

IV. Chakraborty, G. and Sengupta, P.R. (1992) – Proc. Int. AMSE Conf. Calcutta(India), 4, 171.

V. Sengupta, P.R. and PANJA, S. (1991)- Proc.Math. Soc. B.H.U.7, 41.


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M.A.K. Azad, M.A.Aziz, M.S.Alam Sarker



Following Deissler`s approach, the decay of MHD turbulence at times before the final period for the concentrantion fluotuations of a dilute contaminant undergoing a first order chemical reaction in a rotating system is studied. Here two and three point correlatoins between fluctuating quantitles have been third order correlations. The correlations equations are converted to spectrum over all wave numbers, the solution is obtained and this solution gives the Decay law of magnetic energy for the concentration fluctuations before the final period in a rotating system.


MHD turbulence, ,concentration fluctuations, ,energy spectrum, ,magnetic energy,


I. Deissler, R.G.(1958): On the decay of homogeneous turbulence before the final period, Phys.Fluid,1,111-121.

II. Deissler, R.G.(1960): Decay law of homogeneous turbulence for time before the final period., Phys.,Fluid, 3, 176-187.

III. Loefter, A.L and Deisssler, R.G.(1961): Decay of temperature flucations in homogeneous turbluence before the final period, Int. J. Heat mass Transfer, 1,, 312-324.

IV. Kumar, P. and Patel, S.R.(1974): First order reactant in homogeneous turbulence before the final period for the case of multi-point and single time, Physics of fluids, 14, 1362.

V. Kumar, P. and Patel, S.R.(1975): First order reactant in homogeneous turbulence before the final period for the case of multi-point and multi-time, Int.Engng.Sci, 13,305-315.

VI. Patel, S.R.(1976): First order reactant in homogeneous turbulence Numberical results, Int. J.Engng.Sci, 14, 75.

VII. Sarker, M.S.A and Kishor,N(1991): Decay of MHD turbulence brfore thr final prriod, Int.J.Engng,Sci., 29, 1479-485.

VIII. Chandrasekher, S. (1951):  Proc,Roy,Soc., London, A204, 435-449.

IX. Sarker,M.S.A and Islam,M.A(2001): Decay of MHD turbulence brfore thr final prriod for the case of multi-point and multi-time, Indian J.Pure appl. Math,32(7), 1065-1076.

X. Islam,M.A and Sarker,M.A.S (2001): First order reactant in MHD brfore thr final prriod of delay for the case of multi-point and multi-time, Indian J.Pure appl. Math,32(8), 1173-1184.

XI. Sarker,M.S.A and Islam, M.A(2001): Decay of dusyt fluid turbulence brfore thr final prriod in a rotating system, J.Math and Math. Sci, 16, 35-48.

XII. Corrsin, S. (1951b): J. Applied physics, 22,469.

XIII. Sarker,M.S.A and AAzad M.A.K. (2004): Decay of MHD turbulence brfore thr final prriod for the case of multi-point and multi-time in a rotating system, Rajshahi University Studies, Part-B vol.32, 177-192.

XIV. Dixit, T. and Upadhyay, B.N (1989a): Astrophysics and Space Sci.153, 257.

XV. Funada, T., Tuitiya, Y. and Ohji, M. (1978): J.Phy.Soc., Japan, 44 1020.

XVI. Kishor, N. and Dixit, T.()1979: J. Sci.Rce., B.H.U., 30(2),305.

XVII. Kishor, N. and Singh, S.R. (1984): Astrophysics and Space Sci., 104, 121.

XVIII. Kishor, N. and Golsefid, Y.T. ()1988: Astrophysics and Space Sci., 105, 89.

XIX. Kishor, N. and Sarker, M.S.A.(1990b): Astrophysics and Space Sci., 172,279.

XX. Sarker, M.S.A (1998): Rajshani Univ. Studies Part-B, in press.

XXI. Sarker, M.S.A. and Islam,M.A. (2001): Ph.D.Thesis, Dept. of Mathematics, R.U., 56-70.



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M.Ali Akbar, Anup Kumar Datta, Md.Eliyas Karim



A fourth order nonlinear differention equation modeling an over-damped symmetrical system is considered. A perturbation technique is developed in this artical for obtaining the transient responsewhen the eigenvalues are in integral multiple. The results obtained by the presented technique agree with those results obtained by the numerical method nicely. An example is solved to illustrated method.


over-damped symmetrical system ,transient response,forth order non-linear differential equation,eigen values,


I. Ali Akber, M., A.C. Paul and M.A. Sattar, An Asymptotic Method of krylov-Bogoliubov for fourth order over-damped nonlinear system, Geint, J. Bangladesh Math. Soc., vol.22, pp. 83-96, 2002.

II. All  Akbar, M.M. Shamsul Alam and M.A. Sattar, Asymptotic Method for Forth order Damped Nonliner System, Ganit, J. Bangladesh Math. Soc., vol.23, pp. 83-96, 2002.

III. Ali Akbar, M.M. Shamsul Alam and M.A. Satter,  A Simple Technique for Obtaining Certain Over-damped Solutions of an n-th Order Nonlinear Differential Equation, Soochow Journal of Mathematics vol.31(2), pp.291-299, 2005.

IV. Bogoliubov, N.N. and Yu. Mitropolskii, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordan and Breach, New York, 1961.

V. Krylov, N.N. and N.N. Bogoliubov, Introduction to Nonlinear Mechanics, Princeton University Press, New Jersey, 1947.

VI. Mulholland, R.J., Nonlinear Oscillations of Third Order Differential Equation, Int.J. Nonlinear Mechanics, vol.6, pp.279-294, 1971.

VII. Murty, I.S.N., B.L. Deekshatulu and G.Krishna, on an Asymptotic Method of Krylov-Bogoliubov for Over-damped Nonlinear Systems, J.Frank. Inst., vol.288, pp.49-65, 1969.

VIII. Murty, I.S.N., A Unified Krylov-Bogoliubov Method for Solving second Order Nonlinear Systems, Int.J. Nonlinear Mech. Vol.6, pp.45-53, 1971.

IX. Osiniskii. Z., Vibration of a one degree Freedom system with Nonlinear Internal Friction and Rrlaxation, Proceedings of intermations Symposium of Nonlinear Vibrations, vio.111, pp. , . 314-325 Kiew, Lazst, Akand, Nauk Ukr. SSR, 1963.

X. Popov, I.P., A Generalization of the Bogoliubov Asymptotic Method in the Theory of Nonlinear Oscillations (in Russian), Doll. Akad. USSR Vol.3, pp.308-310, 1956.

XI. Sattar, M.A., An asymmetric method for second Order Critically Damped Nonlinear Equations, J.Frank.Inst., vol. 321, pp.109-113, 1986.

XII. Sattar, M.A., An Asymptotic Method for Three-dimensional Over-damped Nonlinear Systems, Habit, J. Bangladesh Math. Soc., Vol.13, pp.1-8, 1993.

XII. Shamsul Alam, M., Asymptotic Methods for second-order Over-damped and Critically Damped Nonlinear Systems, Soochow J. Of Math., Vol.27, pp.187-200, 2001.

XIII. Sattar, M.A., An Asymptotic Method for Three-dimensional Over-damped Nonlinear Systems, Ganit, J. Bangladesh Math. Soc., Vol.13, pp.1-8, 1993.

XIV. Shamsul Alam, M., A Unified Krylov-Bogoliubov-Mitropolskii Method for Solving n-th Order Nonlinear Systems, J.Frank.Inst., vol.339, pp. 239-248, 2002.

XV.Shamsul Alam,M., Some special conditions of third order Over-damped Nonlinear Systems, Indian J. Pure APPL. Math., Vol. 33, pp.727-742, 2002.


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M.Zulfiker Ali, M.Asaduzzaman



In this paper, we find a more generalized contractive mapping that is applied to prove some convergence theorems of Mann Iteration Procedure. Our proof is comparatively easy. Actually, here we generalized some theorems of Rhoades(3), Qihou(1). Ganguly and Bandyopadhy(8) Kannan(12) to develop the concept on convergence of Mann Iteration Procedure.


Contractive mapping,Convergence theorems,Mann Iteration procedure,


I. Qihou, L.: The convergence theorems of the sequence of Ishikawa iterates for quasi convergence mapping, J.Math.Anal.Appl. 146(1990), 301-305.

II. Kalishankar Tiwary and S.C. Debnath: On Ishikawa Iterations, Indian J.Pure and Appl.Math., 26(8) (1995). 743-750.

III. B.E. Rhoades: Some fixed point Iterations , Soochow J.Math. 19(1993), 377-80.

IV. B.E. Rhoades: A General Principle for Mann Iterations, Indian J. Pure and APPL.Math,  26(8)(1995), 751-762.

V. B.E. Rhoades: A comparison of contractive definitions, Trams. Amer.Math. Soc.b(1977), 257-290.

VI.B.E. Rhoades: Some fixed point Iteration procedures, Internet, J.Math.& Math. Sci. vol.14, no.1,(1991), 1-16.

VII. M.K. Chakraborty and B.K.Lahiri: Indian J.Math. 18(1976),81-89.

VIII. D.K.Ganguly and D. Bandyepadhyay: Soochow J.Math. 17(1991), 269-285.

IX. Ishikawa, S.: Fixed point by new Iteration method, Proc. Amer. Meth. Soc.44(1974), 147-150.

X.Ishikawa, S.: Fixed point and Iteration of a non-expansive mapping in a Banach space, Prof.Amar.Math.Soc.59(1976), 65-71.

XI. W.R. Mann: value methods in Iterations, Proc. Amer. Math. Soc. 4(1953), 506-510.

XII. R. Kannan: Construction of fixed points of class of Nonlinear mapping, J. Math. Anal. APPL. 41(1973), 430-438.

XIII. M.M.Vainberg: Variational Method of Monotone Operators in the Theory of Nonlinear Equations, John Wiley & Sons (1973).

XIV. V. Berinde: On the convergence of the Ishikawa Iteration in the class of quasi-contractive operators, Acta. Math. Univ. Comeniance, vol. LXXIII,1 (2004), 119-126.

XV. Arif Rafiq: A convergence theorems for Mann fixed point Iteration procedure, Applied Mathematics E-Notes, 6(2006), 289-293.

XVI. Krishna Kumar: The equivalence of Mann and Ishikawa Iteration for the class of uniformly pseudo-contraction, Thai. J. Math. Vol. 2(2004), no.2, 217-225.

XVII. D.R.Smart: Fixed point theorems, Cambridge University Press (1974).

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A.De, S.Banik



The aim of the present paper is to investigate rotatory vibration of an isotropic  inhomogeneous elastic disk when the elastic constants and also the density of the material varies exponentially as the n-th power of the distance from the center and thinks such problem was not attempted before by any previous investigator and the corresponding results are shown graphically.


Rotator vibration,Isotropic,Inhomogeneous ,Elastic disk,


I. Bhowmick, Monoj Roy & Sengupta, P.R.(1986): The effect of non-homogeneous on rotatory vibration of spherical shell., Indian Journal of Theoretical physics, vol.35, no.4, 1987.

II. Biswas, p.(1983): Nonlinear free vibrations of heated elastic plates. Indian Journal of Pure and Applied Mathematics., 14(10), 1199-1203, October, 1983.

III. Biswas, p.(1983): Nonlinear vibration of a non-isotropic Trapezoidal plate., Indian Journal of Pure and Applied Mathematics,14(2), 265-269, Feb. 1983.

IV. Biswas,p. & Kapoor, (1984):  Nonlinear free vibrations of orthotropic circular plates at elevated temperature. Journal of Indian Institute of Science., 65(B), pp-(87-93), 1984.

V. Chakraborty, A. & Bera, R.K. (2002): Nonlinear vibration and stability of a shallow unsymmetrical orthotropic sandwich shell of double curvature with orthotropic core , Math.Appl., vol.43, pp. (1617-1630), 2002.

VI. Chakraborty,A. & Bera, R.K. (2006):  Large amplitude vibration of thin homogeneous heated orthotropic sandwich elliptic plates. Journal of Thermal stress, vol.29, pp.(21-36), 2006.

VII. Chakraborty, J.G & Choudhuri, P.K.  (1983): The Elastic plastic problem of a thin rotating disk with vibration thickness- Indian Journal of Pure & Applied Mathematics, 14(1), pp-(70-78), Jan 1983.

VIII. Chatterjee, D.(1968): Problem of rotatory vibration of an anisotropic elastic disk. Indian Journal of Mechanics & Mathematics; vol.6 no.1.

IX. Choudhuri P.K. & Dutta, S. (1989): Note on the small vibration of beams with varying Young’s modulus carrying a concentrated mass distribution.’ Indian Journal of Pure & Applied Mathematics, 20(1), 75-88, Jan-1989.

X. De, P.K. (1968): ‘Rotatory vibration of a sphere of variable modulus of elasticity.’  Bulletin of Calcutta Mathematics Society, vol. 60, no.4, p-(185-189).

XI. Kundu, J.C. & Basuli, S(1977): ‘Note on the vibration of circular plates of variable thickness.’ Indian Journal of Theoretical physics; vol.25, no.1, 1977.

XII. Mollah, S.A.(1977): Note on the vibration of circular plates of variable thickness.’ Indian Journal of Theoretical physics; Calcutta;  vol.25, no.1, 1977.

XIII.Mollah, S.A.(1978): ” Vibration of rectangular plates of variable thickness under the combined action of uniform lateral loads and uniform tension’ Bulletin of Calcutta Mathematics Society; vol.70. no.4

XIV. Sinharoy, G.C. & Bera, R.K.(1993): Large amplitude vibration of thin homogeneous orthotropic elastic plates under uniform heating revisited.’ International Journal of Engineering Science, vol-31, no.6, pp.883-892.1993.

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