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Dehghan, Mehdi; Shakeri, Fatemeh

Solution of parabolic integro-differential equations arising in heat conduction in materials with memory via He’s variational iteration technique. (English) Zbl 1192.65158

Int. J. Numer. Methods Biomed. Eng. 26, No. 6, 705-715 (2010).
Summary: We present the solution of some parabolic integro-differential equations, which naturally arise in many applications. He’s variational iteration method is implemented to give the solution for this equation. This technique is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. Application of variational iteration technique to this problem shows that it performs extremely well in terms of accuracy, efficiently, simplicity, stability and reliability.

Cited in 28 Documents

MSC:

65R20 Numerical methods for integral equations

Keywords:

parabolic integro-differential equation; variational iteration method
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\textit{M. Dehghan} and \textit{F. Shakeri}, Int. J. Numer. Methods Biomed. Eng. 26, No. 6, 705--715 (2010; Zbl 1192.65158)
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References:

[1] Ewing, A numerical approximation of non-Fickian flows with mixing length growth in porous media, Acta Mathematica Universitatis Comenianae 70 pp 75– (2001) · Zbl 1007.76036
[2] Ganji, Application of He’s variational iteration method to nonlinear Jaulent Miodek equations and comparing it with ADM, Journal of Computational and Applied Mathematics 207 pp 35– (2007) · Zbl 1120.65107
[3] Tian, Shockâpeakon and shockâcompacton solutions for K(p, q) equation by variational iteration method, Journal of Computational and Applied Mathematics 207 pp 46– (2007)
[4] Khuri, A new approach to Bratu’s problem, Applied Mathematics and Computation 147 (1) pp 131– (2004) · Zbl 1032.65084
[5] Dehghan, Application of He’s variational iteration method for solving the Cauchy reactionâdiffusion problem, Journal of Computational and Applied Mathematics 214 pp 435– (2008) · Zbl 1135.65381
[6] He, Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons and Fractals 29 pp 108– (2006) · Zbl 1147.35338
[7] Lu, Variational iteration method for solving two-point boundary value problems, Journal of Computational and Applied Mathematics 207 pp 92– (2007) · Zbl 1119.65068
[8] Sweilam, Variational iteration method for solving cubic nonlinear Schrodinger equation, Journal of Computational and Applied Mathematics 207 pp 155– (2007) · Zbl 1119.65098
[9] Momani, Numerical solutions of the space-time fractional advectionâdispersion equation, Numerical Methods for Partial Differential Equations
[10] Odibat, Numerical methods for nonlinear partial differential equations of fractional order, Applied Mathematical Modelling 32 pp 28– (2008) · Zbl 1133.65116
[11] Dehghan, Identifying an unknown function in a parabolic equation with overspecified data via He’s variational iteration method, Chaos, Solitons and Fractals 36 pp 157– (2008) · Zbl 1152.35390
[12] Alia, Variational iteration method for solving partial differential equations with variable coefficients, Chaos, Solitons and Fractals
[13] Dehghan, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy 13 pp 53– (2008)
[14] Alia, Variational iteration method for solving biharmonic equations, Physics Letters A 370 pp 441– (2007)
[15] Shakeri, Numerical solution of a biological population model using He’s variational iteration method, Computers and Mathematics with Applications 54 pp 1197– (2007) · Zbl 1137.92033
[16] Wazwaz, The variational iteration method for rational solutions for KdV, K(2, 2), Burgers, and cubic Boussinesq equations, Journal of Computational and Applied Mathematics 207 pp 18– (2007) · Zbl 1119.65102
[17] Dehghan, The use of He’s variational iteration method for solving the FokkerâPlanck equation, Physica Scripta 74 pp 310– (2006) · Zbl 1108.82033
[18] Abbasbandy, Numerical method for non-linear wave and diffusion equations by the variational iteration method, International Journal for Numerical Methods in Engineering 73 pp 1836– (2008) · Zbl 1159.76372
[19] Geng, Numerical solution of a system of fourth order boundary value problems using variational iteration method, Applied Mathematics and Computation 200 pp 231– (2008) · Zbl 1153.65348
[20] Shakeri, Numerical solution of the KleinâGordon equation via He’s variational iteration method, Nonlinear Dynamics 51 pp 89– (2008) · Zbl 1179.81064
[21] Abbasbandy, An approximation solution of a nonlinear equation with Riemann Liouville’s fractional derivatives by He’s variational iteration method, Journal of Computational and Applied Mathematics 207 pp 53– (2007) · Zbl 1120.65133
[22] Sweilam, Harmonic wave generation in non linear thermoelasticity by variational iteration method and Adomian’s method, Journal of Computational and Applied Mathematics 207 pp 64– (2007) · Zbl 1115.74028
[23] He, Variational iteration method for delay differential equations, Communications in Nonlinear Science and Numerical Simulation 2 (4) pp 235– (1997) · Zbl 0924.34063
[24] He, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B 20 pp 1141– (2006) · Zbl 1102.34039
[25] He, Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation 114 pp 115– (2000) · Zbl 1027.34009
[26] He, Approximate analytical solution of Blasius’ equation, Communications in Nonlinear Science and Numerical Simulation 4 (1) pp 75– (1999) · Zbl 0932.34005
[27] Batiha, Application of variational iteration method to the generalized BurgersâHuxley equation, Chaos, Solitons and Fractals 36 pp 660– (2008)
[28] Javidi, Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method, Chaos, Solitons and Fractals 36 pp 309– (2008) · Zbl 1350.35182
[29] Abassy, Toward a modified variational iteration method, Journal of Computational and Applied Mathematics 207 pp 137– (2007) · Zbl 1119.65096
[30] Abassy, Solving nonlinear partial differential equations using the modified variational iteration Padé technique, Journal of Computational and Applied Mathematics 207 pp 73– (2007) · Zbl 1119.65095
[31] Inc, Numerical simulation of KdV and mKdV equations with initial conditions by the variational iteration method, Chaos, Solitons and Fractals 34 pp 1075– (2007) · Zbl 1142.35572
[32] Inc, Exact special solutions to the nonlinear dispersive K(2, 2, 1) and K(3, 3, 1) equations by He’s variational iteration method, Nonlinear, Analysis: Theory, Methods and Applications 69 pp 624– (2008) · Zbl 1159.35413
[33] Inc, Numerical doubly-periodic solution of the (2+1)-dimensional Boussinesq equation with initial conditions by the variational iteration method, Physics Letters A 366 pp 20– (2007) · Zbl 1203.65210
[34] Tatari, Solution of problems in calculus of variations via He’s variational iteration method, Physics Letters A 362 pp 401– (2007) · Zbl 1197.65112
[35] Shakeri, Solution of a model describing biological species living together using the variational iteration method, Mathematical and Computer Modelling 48 pp 685– (2008) · Zbl 1156.92332
[36] Tatari, He’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation, Chaos, Solitons and Fractals 33 pp 671– (2007) · Zbl 1131.65084
[37] Dehghan, Variational iteration method for solving the wave equation subject to an integral conservation condition, Chaos, Solitons and Fractals (2008)
[38] He, Approximate solution of nonlinear differential equations with convolution product nonlinearities, Computer Methods in Applied Mechanics and Engineering 167 pp 57– (1998) · Zbl 0932.65143
[39] He, A new approach to nonlinear partial differential equations, Communications in Nonlinear Science and Numerical Simulation 2 (4) pp 230– (1997) · Zbl 0923.35046
[40] Tatari, On the convergence of He’s variational iteration method, Journal of Computational and Applied Mathematics 207 pp 121– (2007) · Zbl 1120.65112
[41] Cushman, Nonlocal reactive transport with physical and chemical heterogeneity, Water Resources Research 31 pp 2219– (1995)
[42] Renardy, Mathematical Problems in Viscoelasticity (1987)
[43] Cushman, Nonlocal dispersion in media with continuously evolving scales of heterogeneity, Transport in Porous Media 13 pp 123– (1993)
[44] Dagan, The significance of heterogeneity of evolving scales to transport in porous formations, Water Resource Research 30 pp 3327– (1994)
[45] Dehghan, Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Mathematics and Computers in Simulation 71 pp 16– (2006) · Zbl 1089.65085
[46] Dehghan, Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numerical Methods for Partial Differential Equations 21 pp 611– (2005) · Zbl 1069.65104
[47] Dehghan, Meshless local PetrovâGalerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, Applied Numerical Mathematics (2008)
[48] Shakourifar, On the numerical solution of nonlinear systems of Volterra integro-differential equations with delay arguments, Computing (2008) · Zbl 1154.65098
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.