Investigation of approximate solutions of evolution type partial differential equations in terms of high performance computings

For evolution type partial differential equations, B-Spline collocation method, variational iteration, Laplace variational iteration, Adomian decomposition and Laplace-Adomian decomposition methods are discussed. The Gardner equation, which was solved by the exponential B-Spline collocation method, was solved by the Laplace-Adomian decomposition and Laplace variational iteration method, and the solutions were compared. Various direct and iterative methods are explained for the solution of linear equation systems obtained when the B-spline collocation method is applied. In the obtained linear equation systems in the form of "AX=B", some algorithms used in cases where the matrix A is large-sized and sparse matrix, the necessary libraries, high-performance computation systems and, in parallel, the calculations in the literature for large-sized matrices are given.