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The one-dimensional heat equation subject to a boundary integral specification

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  • Dehghan, Mehdi

Abstract

Various processes in the natural sciences and engineering lead to the nonclassical parabolic initial boundary value problems which involve nonlocal integral terms over the spatial domain. The integral term may appear in the boundary conditions. It is the reason for which such problems gained much attention in recent years, not only in engineering but also in the mathematics community. In this paper the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered. Several approaches for the numerical solution of this boundary value problem which have been considered in the literature, are reported. New finite difference techniques are proposed for the numerical solution of the one-dimensional heat equation subject to the specification of mass. Numerical examples are given at the end of this paper to compare the efficiency of the new techniques. Some specific applications in various engineering models are introduced.

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  • Dehghan, Mehdi, 2007. "The one-dimensional heat equation subject to a boundary integral specification," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 661-675.
    • Handle: RePEc:eee:chsofr:v:32:y:2007:i:2:p:661-675
    DOI: 10.1016/j.chaos.2005.11.010
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    References listed on IDEAS

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    1. Abdelfatah Bouziani, 2002. "On the solvability of parabolic and hyperbolic problems with a boundary integral condition," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 31, pages 1-13, January.
    2. Abdelfatah Bouziani, 2003. "On the weak solution of a three-point boundary value problem for a class of parabolic equations with energy specification," Abstract and Applied Analysis, Hindawi, vol. 2003, pages 1-17, January.
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    Cited by:

    1. Dehghan, Mehdi & Saadatmandi, Abbas, 2009. "Variational iteration method for solving the wave equation subject to an integral conservation condition," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1448-1453.
    2. Dehghan, Mehdi & Shakourifar, Mohammad & Hamidi, Asgar, 2009. "The solution of linear and nonlinear systems of Volterra functional equations using Adomian–Pade technique," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2509-2521.
    3. Deng, Aimin & Lin, Ji & Liu, Chein-Shan, 2022. "Boundary shape function iterative method for nonlinear second-order boundary value problems with nonlinear boundary conditions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 539-551.
    4. Cui, Ming Rong, 2015. "Convergence analysis of compact difference schemes for diffusion equation with nonlocal boundary conditions," Applied Mathematics and Computation, Elsevier, vol. 260(C), pages 227-241.
    5. Yüzbaşı, Şuayip, 2018. "A collocation approach for solving two-dimensional second-order linear hyperbolic equations," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 101-114.
    6. Martín-Vaquero, J., 2009. "Two-level fourth-order explicit schemes for diffusion equations subject to boundary integral specifications," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2364-2372.

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