A numerical solution of an inverse diffusion problem based on operational matrices of orthonormal polynomials

Abstract

The inverse problem of identifying the diffusion coefficient in the one‐dimensional parabolic heat equation is studied. We assume that the information of Dirichlet boundary conditions along with an integral overdetermination condition is available. By applying the given assumptions, the problem is reformulated as a nonclassical parabolic equation along with the initial and boundary conditions. Then we employ the direct technique based on the operational matrices for integration, differentiation, and the product of the orthonormal polynomials together with the Ritz–Galerkin technique to reduce the main problem to the solution of a system of nonlinear algebraic equations. Numerical simulations are presented to show the applicability of the proposed method.

Reconstruction of a time‐dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method

“…[17][18][19] In the present contribution, we use a spectral technique [53][54][55][56][57][58][59] which is a powerful tool to provide more accurate and at the same time stable numerical solution for the inverse problems (1.1)- (1.5). By applying the satisfier function 21,[60][61][62][63][64][65] which fulfills all the initial and boundary conditions and employing this function within a linear transformation, we get an equivalent problem with the homogeneous initial and boundary conditions. We expand the desired unknown function in terms of orthonormal Bernstein basis functions (OBBFs); the operational matrices 21,[66][67][68][69][70][71][72] of integration and differentiation of these polynomials are utilized to formulate the discrete version of the problem.…”

“…where P N and D N are called the (N +1)×(N +1) operational matrices of integration and differentiation, respectively. 21,71,72 By considering the entries of these matrices as p i𝑗 , d i𝑗 , i, 𝑗 = 0, N and…”

“…By applying the satisfier function 21,[60][61][62][63][64][65] which fulfills all the initial and boundary conditions and employing this function within a linear transformation, we get an equivalent problem with the homogeneous initial and boundary conditions. We expand the desired unknown function in terms of orthonormal Bernstein basis functions (OBBFs); the operational matrices 21,[66][67][68][69][70][71][72] of integration and differentiation of these polynomials are utilized to formulate the discrete version of the problem. Finally, the main problem is reduced to the solution of a system of nonlinear algebraic equations which is done by employing Newton's iteration method.…”

“…Moreover, if the property of the medium under study does not change rapidly, the unknown coefficient can be space‐wise dependent solely 20 . However, in the general form, it depends on the solution of the direct problem 7,21 …”

Recovery of coefficients of a heat equation by Ritz collocation method

“…Moreover, if the property of the medium under study does not change rapidly, the unknown coefficient can be space-wise dependent solely (Liao, 2011). However, in the general form it depends on the solution of the direct problem (Rashedi, 2021;Samarskii & Vabishchevich, 2008).…”