A Novel Approach for Solving an Inverse Reaction–Diffusion–Convection Problem

In this paper, we consider an inverse reaction–diffusion–convection problem in which one of the boundary conditions is unknown. A sixth-kind Chebyshev collocation method will be proposed to solve numerically this problem and to obtain the unknown boundary function. Since this inverse problem is generally ill-posed, to find an optimal stable solution, we will utilize a regularization method based on the mollification technique with the generalized cross-validation criterion. The error estimate of the numerical solution is investigated. Finally, to authenticate the validity and effectiveness of the proposed algorithm, some numerical test problems are presented.

[1]  Diego A. Murio,et al.  Time fractional IHCP with Caputo fractional derivatives , 2008, Comput. Math. Appl..

[2]  Yongzhi S. Xu,et al.  Ritz–Galerkin method for solving an inverse problem of parabolic equation with moving boundaries and integral condition , 2019 .

[3]  Luca Zanni,et al.  Inverse problems in machine learning: An application to brain activity interpretation , 2008 .

[4]  A. I. Prilepko,et al.  Methods for solving inverse problems in mathematical physics , 2000 .

[5]  A basic class of symmetric orthogonal polynomials using the extended Sturm–Liouville theorem for symmetric functions , 2007, 1305.5669.

[6]  Afshin Babaei,et al.  Reconstructing unknown nonlinear boundary conditions in a time‐fractional inverse reaction–diffusion–convection problem , 2018, Numerical Methods for Partial Differential Equations.

[7]  F. Marcuzzi,et al.  A parabolic inverse convection-diffusion-reaction problem solved using space-time localization and adaptivity , 2013, Appl. Math. Comput..

[8]  Afshin Babaei,et al.  A Stable Numerical Approach to Solve a Time-Fractional Inverse Heat Conduction Problem , 2018 .

[9]  S. Arridge,et al.  Optical tomography: forward and inverse problems , 2009, 0907.2586.

[10]  A. Babaei,et al.  The Sinc‐Galerkin method for solving an inverse parabolic problem with unknown source term , 2013 .

[11]  Diego A. Murio,et al.  Stable numerical solution of a fractional-diffusion inverse heat conduction problem , 2007, Comput. Math. Appl..

[12]  A. Baklanov,et al.  Direct and Inverse Problems in a Variational Concept of Environmental Modeling , 2012, Pure and Applied Geophysics.

[13]  Youssri Hassan Youssri,et al.  Sixth-Kind Chebyshev Spectral Approach for Solving Fractional Differential Equations , 2019, International Journal of Nonlinear Sciences and Numerical Simulation.

[14]  Hatef Dastour,et al.  Estimation of unknown boundary functionsin an inverse heat conduction problem using a mollified marching scheme , 2014, Numerical Algorithms.

[15]  Dilip N. Ghosh Roy,et al.  Inverse Problems and Inverse Scattering of Plane Waves , 2001 .

[16]  Keith A. Woodbury,et al.  Inverse Engineering Handbook , 2002 .

[17]  M. Abdelkawy,et al.  Efficient Spectral Collocation Algorithm for Solving Parabolic Inverse Problems , 2016 .

[18]  Liu Yang,et al.  An Optimal Control Method for Nonlinear Inverse Diffusion Coefficient Problem , 2014, J. Optim. Theory Appl..

[19]  V. Isakov.  Appendix -- Function Spaces , 2017 .

[20]  Malihe Rostamian,et al.  A meshless method to the numerical solution of an inverse reaction–diffusion–convection problem , 2017, Int. J. Comput. Math..