AN EFFICIENT NUMERICAL TECHNIQUE FOR SOLVING HEAT EQUATION WITH NONLOCAL BOUNDARY CONDITIONS

Yıl 2022, Cilt: 6 Sayı: 2, 157 - 167, 30.06.2022

Zakia Hammouch Anam Zahra Azız Rehman Syed Ali Mardan

https://doi.org/10.31197/atnaa.846217

Cited By: 1

Öz

A third order parallel algorithm is proposed to solve one dimensional non-homogenous heat equation with integral boundary conditions. For this purpose, we approximate the space derivative by third order finite difference approximation. This parallel splitting technique is combined with Simpson's 1/3 rule to tackle the nonlocal part of this problem. The algorithm develop here is tested on two model problems. We conclude that our method provides better accuracy due to availability of real arithmetic.

Anahtar Kelimeler

Parabolic partial differential equation, Non-local boundary conditions, Finite difference scheme, Integral boundary condition.

Kaynakça

• [1] S. Abbasbandy, B. Soltanalizadeh, http://dx.doi.org/10.1080/00207160.2010.521816, A matrix formulation to the wave equation with non-local boundary condition, International Journal of Computational Mathematics, 88(2011).
• [2] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, https://doi.org/10.1002/mma.6652, On the solution of a boundary value problem associated with a fractional differential equation, Mathematical Methods in the Applied Sciences, (2020).
• [3] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On The Solutions Of Fractional Differential Equations Via Geraghty Type Hybrid Contractions, Applied Computation Mathematics, 20(2021).
• [4] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, https://doi.org/10.1007/s13398-021-01095-3, Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions, RACSAM 115(2021).
• [5] S. Afshar, B. Soltanalizadeh, http://www.ijpam.eu/contents/2014-94-2/1/1.pdf, Solution of the two-dimensional second- order diffusion equation with nonlocal boundary condition, International Journal of Pure Applied Mathematics, 94(2014), 119-131.
• [6] M. Asif, et al., Legendre multi-wavelets collocation method for numerical solution of linear and nonlinear integral equations, Alexandria Engineering Journal, 59.6(2020).
• [7] Babakhani, Azizollah, and Qasem Al-Mdallal, On the existence of positive solutions for a non-autonomous fractional differential equation with integral boundary conditions, Computational Methods for Di?erential Equations (2020).
• [8] J.G. Batten, http://dx.doi.org/10.12732/ijpam.v94i2.1, Solution Of the two-dimensional second-Order diffusion equation with nonlocal boundary condition, International Journal of Pure Applied Mathematics, 94(2014).
• [9] J.R. Cannon, https://www.researchgate.net/publication/295869412, A new numerical method for heat equation subject to integral specifications, Quart. Appl. Math., no. 21(1963).
• [10] M. Dehghan, https://doi.org/10.1016/S0096-3003(02)00479-4, Numerical solution of a parabolic equation with non-local boundary specifications, Applied Mathematics and Computation, 145(2003).
• [11] M. Dehghan, http://dx.doi.org/10.1080/00207160500069847, Solution of a partial integro-differential equation arising from viscoelasticity, International Journal of Computer Mathematics, 83(2006).
• [12] A.B. Gumel, https://doi.org/10.1017/S0334270000010560, On the numerical solution of the diffusion equation subject to the specification of mass, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics 40(1999).
• [13] L.I. Kamynin , http://www.sciencedirect.com/sdfe/pdf/download/eid/1-s2.0-0041555364900801/first-page-pdf, A bound- ary value problem in the theory of heat conduction with a nonclassical boundary condition, Zh. Vych. Math., 4(1962).
• [14] J.E. Lazreg, S. Abbas, M. Benchohra, and E. Karapinar, Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces, Open Mathematics, 19(2021).
• [15] Al-Mdallal, Qasem M, Monotone iterative sequences for nonlinear integro-differential equations of second order, Nonlinear Analysis: Real World Applications, 12.(2011).
• [16] M.A. Rehman, M.S. A Taj, https://www.researchgate.net/publication/266606421, Fourth order method for non homoge- neous heat equation with nonlocal boundary conditions, Applied Mathematical Sciences, 3(2009).
• [17] M.A. Rehman, M.S. A Taj, https://www.researchgate.net/publication/266606421, Fourth order method for non homoge- neous heat equation with nonlocal boundary conditions, Applied Mathematical Sciences, 3(2009).
• [18] A. Salim, B. Benchohra, E. Karapinar, J.E. Lazreg, Existence and Ulam stability for Hilfer-type fractional differential equations, Advance Differential Equation, 601(2020).
• [19] B. Soltanalizadeh, https://www.researchgate.net/publication/285991927, A numerical method to the one dimential heat equation with an integral condition, Australian Journal of Basic and Applied Sciences, 10(2010).
• [20] M.S.A Taj and E.H. Twizell,http://dx.doi.org/10.1080/00207169808804672, A family of third-order parallel splitting meth- ods for parabolic partial di?erential equations, International Journal of Computational Mathematics, 3(1996).
• [21] M.S.A Taj and E.H. Twizell, http://dx.doi.org/10.1002/(SICI)1098-2426(199707)13:4<357::AID-NUM4>3.0.CO;2-K, A family of fourth-order parallel splitting methods for parabolic partial di?erential equations, Numerical Methods for partial di?erential equations, 4(1998).
• [22] M.S.A Taj and E.H. Twizell, http://dx.doi.org/10.1080/00207169808804672, A family of third-order parallel splitting methods for parabolic partial differential equations, International Journal of Computational Mathematics, 67(1998).
• [23] M.S.A Taj and M.A Rehman, Fifth order numerical method for heat equation with nonlocal boundary conditions, Math- ematical computational sciences, 4(2014).
• [24] S.A Mardan and M.A Rehman, Fusion higher order parallel splitting methods for parabolic partial di?erential equations, International mathematical forum, 7(2012).