The problem of recovering a time-dependent coefficient in a parabolic partial differential equation has attracted considerable attention in the classical literature on inverse heat conduction problems. In this article, the Ritz-Galerkin method with satisfier function is utilized to solve the inverse problems of identifying the temperature u.x; t / and the unknown diffusion coefficient a.t / in the one-dimensional heat equation from various additional information. Numerical examples are presented and discussed.

SummaryIn this article, a new numerical method based on Bernoulli wavelet basis has been applied to give the approximate solution of the fractional optimal control problems. The new operational matrices of multiplication and fractional integration are constructed. The proposed method is applied to reduce the problem to the solution of a system of algebraic equations. The fractional derivative is considered in the Caputo sense. Convergence of the algorithm is proved and some results concerning the error analysis are obtained. Approximate solutions are given and in the cases when we have an exact solution, a comparison with the exact solution is presented to demonstrate the validity and applicability of the proposed method. In addition, we compare the obtained results with the results of other methods. Comparison shows the more accuracy of presented technique in comparison to other published methods.

Purpose -The purpose of this paper is to implement the Ritz-Galerkin method in Bernstein polynomial basis to give approximation solution of a parabolic partial differential equation with non-local boundary conditions. Design/methodology/approach -The properties of Bernstein polynomial and Ritz-Galerkin method are first presented, then the Ritz-Galerkin method is utilized to reduce the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique. Findings -The authors applied the method presented in this paper and solved three test problems. Originality/value -This is the first time that the Ritz-Galerkin method in Bernstein polynomial basis is employed to solve the model investigated in the current paper.

In this article, a new method is introduced for finding the exact solution of the product form of parabolic equation with nonlocal boundary conditions. Approximation solution of the present problem is implemented by the Ritz-Galerkin method in Bernoulli polynomials basis. The properties of Bernoulli polynomials are first presented, then Ritz-Galerkin method in Bernoulli polynomials is used to reduce the given differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the techniques presented in this article for finding the exact and approximation solutions.

In this paper, the two-dimensional Bernoulli wavelets (BWs) with Ritz-Galerkin method are applied for the numerical solution of the time fractional diffusionwave equation. In this way, a satisfier function which satisfies all the initial and boundary conditions is derived. The two-dimensional BWs and Ritz-Galerkin method with satisfier function are used to transform the problem under consideration into a linear system of algebraic equations. The proposed scheme is applied for numerical solution of some examples. It has high accuracy in computation that leads to obtaining the exact solutions in some cases.