A new approach for determination of boundary function for diffusion equation by fourier method

A new approach for determination of boundary function for diffusion equation by fourier method

We consider a linear diffusion equation with a nonlocal boundary condition. We attempt to recover the boundary condition and the solution of diffusion equation for a problem by making use of an over-determination condition of integral type. Explicit solutions for these unknowns are derived by employing Fourier method. We obtain sufficient conditions for the existence and uniqueness of the solution and determination of boundary condition.

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  • D. Mantzavinos, A.S. Fokas, The unified method for the heat equation: I. non-separable boundary conditions and non-local constraints in one dimension, Euro. J. Appl. Math. 24, (2013), 857–886.
  • E. K. Lenzi, H. V. Ribeiro, J. Martins, M. K. Lenzi, G. G. Lenzi, S. Specchia; Non-Markovian diffusion equation and diffusion in a porous catalyst, Chemical Engineering Journal 172, (2011), 1083–1087.
  • G. Freiling, V.A. Yurko, Inverse problems for Sturm–Liouville differential operators with a constant delay, Appl. Math. Lett. 25, (2012),1999–2004.
  • G. Özkum, A. Demir, S. Erman, E. Korkmaz, B. Özgür, On the Inverse Problem of the Fractional Heat-Like Partial Differential Equations: Determination of the Source Function, Advances in Mathematical Physics 2013,(2013), 1–8.
  • G. Wei, X. Wei, A generalization of three spectra theorem for inverse Sturm-Liouville problems, App. Math. Lett. 35, (2014) 41-45.
  • J. Cannon, Determination of an unknown heat source from overspecified boundary data, SIAM J. Numer. Anal. 2, (1968), 275–286.
  • J. R. Cannon, S. P. Esteva, J. V. D. Hoek, A Galerkin procedure for the diffusion subject to the specification of mass, SIAM J. Numer. Anal. 24, No. 3,(1987),499–515.
  • J. R. Cannon, Y. Lin, S. Wang, Determination of a control parameter in a parabolic partial differential equation, J. Austral. Math. Soc. Ser. B. 33,(1991),149–163.
  • K.M. Levere, H. Kunze, D. Torre, A collage-based approach to solving inverse problems for second-order nonlinear parabolic PDEs, J. Math. Anal. Appl. 406 ,no. 1,(2013) , 120–133.
  • L.O.C. Pérez, J.P. Zubell, On the Inverse Problem for Scattering of Electromagnetic Radiation by a Periodic Structure, Studies in Applied Mathematics 111,(2003), 115–166.
  • M.C. Drignei, Uniqueness of solutions to inverse Sturm–Liouville problems with L^2 (0,a) image potential using three spectra, Adv. in Appl. Math. 42, (2009), 471–482.
  • M. Kirane, S.A. Malik, and M. A. Al-Gwaiz, An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions, Math. Meth. Appl. Sci. 36, (2013), 1056–1069.
  • M. Kirane, S.A. Malik, Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time, Appl. Math.Comp. 218, (2011), 163–170.
  • M. Mierzwiczak, J.A. Kolodziej, Application of the method of fundamental solutions and radial basis functions for inverse transient heat source problem , Comput. Phys. Commun. 12, (2010), 2035–2043.
  • P. DuChateau, An introduction to inverse problem in partial differential equations for engineers, physicists and mathematicians, In Proceedings of workshop on parameter identification and inverse problems in hydrology, geology and ecology. Edited by J. Gottlieb, P. DuChateau, Kluwer Academic Publishers, The Netherlands,(1995), 3–38.
  • P.J. Bendeich, J.M. Barry, W. Payten, Determination of specific heat with a simple inverse approach, Appl. Math. Modelling. 27, (2003), 337–344.
  • R. Zolfaghari, A. Shidfar, Reconstructing an unknown time-dependent function in the boundary conditions of a parabolic PDE, Appl. Math. Comput. 226, (2014), 238–249.
  • S. Wang, M. Zhao, X. Li, Radial anomalous diffusion in an annulus, Physica A. 390, (2011), 3397–3403.
  • T.S. Aleroev, M. Kirane, S.A. Malik, Determination of a source term for a time fractional diffusion equation with an integral type over-determining condition, E.J.D.E. 270,(2013), 1–16.
  • V. Yurko, An inverse problem for Sturm–Liouville differential operators on A-graphs, Appl. Math. Lett. 23,(2010),875–879.
  • Y.M. Kwon, W.K.Park, Analysis of subspace migrations in limited-view inverse scattering problems, Appl. Math. Lett. 26, (2013), 1107–1113.
  • Z.C. Deng, L. Yang, N. Chen, Uniqueness and stability of the minimizer for a binary functional arising in an inverse heat conduction problem, J. Math. Anal. Appl. 382, (2011), no. 1 474–486.
  • Z. Sun, Inverse boundary value problems for a class of semilinear elliptic equations, Adv. in Appl. Math. 32, (2004), 791–800.