A wavelet-based collocation technique to find the discontinuous heat source in inverse heat conduction problems

Abstract

This paper is devoted to an inverse problem of determining discontinuous space-wise dependent heat source in a linear parabolic equation from the measurements at the final moment. In the existing literature, a considerable accurate solution to the inverse problems with unknown space-wise dependent heat source is impossible without introducing any type of regularization method but here we have determine the unknown discontinuous space-wise dependent heat source acutely using Haar wavelet collocation method (HWCM) without applying the regularization technique. This HWCM is based on finite-difference and Haar wavelets approximation to the inverse problem. In contrast to other numerical techniques, in HWCM, we use Haar functions that create a well-conditioned system of algebraic equations. The proposed method is stable and convergent because the numerical solution converges to the exact solution without observing any difficulty. Finally, some numerical examples are presented to verify the validity of the HWCM for different cases of source term.

Boubaker Scaling Operational Matrices for Solving Calculus of Variation Problems

“…In addition, the Boubaker wavelets were applied in [20] for the numerical solution of important problems that describe phenomena in mathematical science and astrophysics, namely the thermal explosions and the stellar structure. The Haar wavelets, finite difference method and Crank-Nicolson finite difference method are used in [21][22][23][24] for solving some direct and indirect problems. Further, in [25], the Rayleigh-Ritz method was extended together with operational matrices of different orthogonal polynomials such as Gegenbauer polynomials, shifted Legendre polynomials and shifted Chebyshev polynomials of the first, the third and the fourth kinds to solve a certain class for variational problems.…”