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Numerical solutions to integral equations equivalent to differential equations with fractional time

Bartosz Bandrowski;  Anna Karczewska;  Piotr Rozmej — 2010

International Journal of Applied Mathematics and Computer Science

This paper presents an approximate method of solving the fractional (in the time variable) equation which describes the processes lying between heat and wave behavior. The approximation consists in the application of a finite subspace of an infinite basis in the time variable (Galerkin method) and discretization in space variables. In the final step, a large-scale system of linear equations with a non-symmetric matrix is solved with the use of the iterative GMRES method.

A finite element method for extended KdV equations

Anna Karczewska;  Piotr Rozmej;  Maciej Szczeciński;  Bartosz Boguniewicz — 2016

International Journal of Applied Mathematics and Computer Science

The finite element method (FEM) is applied to obtain numerical solutions to a recently derived nonlinear equation for the shallow water wave problem. A weak formulation and the Petrov-Galerkin method are used. It is shown that the FEM gives a reasonable description of the wave dynamics of soliton waves governed by extended KdV equations. Some new results for several cases of bottom shapes are presented. The numerical scheme presented here is suitable for taking into account stochastic effects, which...

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