Reconstruction of Timewise Term for the Nonlocal Diffusion Equation from an Additional Condition
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RESEARCH PAPER
Reconstruction of Timewise Term for the Nonlocal Diffusion Equation from an Additional Condition M. J. Huntul1
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Mohammad Tamsir1
Received: 20 May 2020 / Accepted: 31 August 2020 Ó Shiraz University 2020
Abstract The inverse problem of reconstructing the timewise term along with the temperature in the nonlocal diffusion equation with initial and boundary conditions and additional measurement (an integral and partial heat flux) is, for the first time, numerically investigated. This inverse problem appears extensively in the modeling of various phenomena in engineering and physics. The inverse problem considered in this paper has a unique solution. A Crank–Nicolson FDM is applied as a direct solver. The resulting nonlinear problem is solved using the MATLAB subroutine to minimize the objective functional. The Tikhonov regularization method is applied where necessary. A pair of examples, with smooth and non-smooth continuous timewise term, are discussed to assess the accuracy and stability of the numerical solutions. Keywords Nonlocal diffusion equation Inverse problem Integral condition Tikhonov regularization Nonlinear optimization
1 Introduction The parameter reconstruction in the parabolic partial differential equation from the over-specified data plays key role in engineering and sciences. These problems are extensively used to find the unknown coefficients (Dehghan 2001, 2005a). These unknown coefficients are essential to the physical process but usually cannot be determined directly (Dehghan 2005b). Dehghan (2003) presented FDM for the inverse problem ut ¼ uxx þ pðtÞu þ /ðx; tÞ, to find the control parameter p(t). The numerical solution of the inverse problem ut ¼ aðtÞuxx with an extra measurement is discussed by several authors (Dehghan and Tatari 2005; Lakestani and Dehghan 2010) to find the thermal conductivity a(t). The inverse problem of determining the thermal diffusivity from boundary data has been examined extensively by many authors in the past, see (Azari et al. 2004; Cannon and Rundell 1991; Hussein et al. 2016; Ivanchov 1998; Jones 1962, 1963) to mention only a few. It is worth to notice that & M. J. Huntul [email protected] 1
Department of Mathematics, College of Science, Jazan University, Jazan, Saudi Arabia
many other types of inverse problem are examined in (ZuiCha et al. 2013; Fatullayev and Cula 2009; Hafid and Lacroix 2016; Nedin et al. 2016. In the recent papers (Huntul and Lesnic 2017, 2020) by the authors, we have investigated the inverse problem of finding the timewise thermal conductivity for one or two-dimensional parabolic equations. Recently, Ivanchov (2012) investigated the inverse problem theoretically and proved the existence and uniqueness conditions. This work examines the inverse R ‘ problem to recover the timewise term a 0 uðx; tÞdx numerically in the rectangular domain with the additional non-local integral and partial heat flux condition. The inverse problem presented in this paper has alread
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