Determination of an unknown function in a parabolic inverse problem by Sinc-collocation method

Abstract

In this article, an inverse problem of determining an unknown time-dependent source term of a parabolic equation is considered. We change the inverse problem to a Volterra integral equation of convolution-type. By using Sinc-collocation method, the resulting integral equation is replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the condition number and the norm of the inverse of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. Some examples are given to demonstrate the computational efficiency of the method.

Cardinal Hermite interpolant multiscaling functions for solving a parabolic inverse problem

“…These examples are chosen such that their analytical solutions are known. Example 1 was considered in [33] by sinc-collocation method, and the absolute value of the errors (AVE) for p(t) and u(x, 0.5) for N 1 = 5, 10, 15 and 20 were given. In [33], the step size in the definition of the sinc-collocation method are appropriately chosen depending on N 1 .…”

“…Example 1 was considered in [33] by sinc-collocation method, and the absolute value of the errors (AVE) for p(t) and u(x, 0.5) for N 1 = 5, 10, 15 and 20 were given. In [33], the step size in the definition of the sinc-collocation method are appropriately chosen depending on N 1 . For this example we report the AVE of our method with J = 1 and J = 2 with [33], for N 1 = 15 and N 1 = 20.…”

“…In [33], the step size in the definition of the sinc-collocation method are appropriately chosen depending on N 1 . For this example we report the AVE of our method with J = 1 and J = 2 with [33], for N 1 = 15 and N 1 = 20. Example 2 was first considered in [15] by using the finite-difference method and also solved in [33].…”

“…In Tables 1 and 2, we compare the absolute error of our method with J = 1 and J = 2 together with the sinc-collocation method presented in [33] for p(t) and v(x, 0.5), respectively.…”

“…In this example to show the stability of our method, consider the following perturbed equation [15,33]: …”