Numerical solutions to integral equations equivalent to differential equations with fractional time. (English) Zbl 1201.35020

This paper presents an approximate method of solving a fractional (in the time variable) equation which describes the processes lying between heat and wave behavior. In most of applications related to fractional differential or fractional integro-differential equations, the numerical methods are limited to \(1+1\) (time + space) dimensions. This paper presents a different method for solving a fractional integro-differential equation which can handle more dimensional cases within a good approximation. The method is limited to cases when the initial condition is smooth enough with respect to space variables \(x\). In such cases the approach works also for \((1+2)\) and \((1+3\)) dimensions. More precisely 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.


35A35 Theoretical approximation in context of PDEs
45D05 Volterra integral equations
65R20 Numerical methods for integral equations
35R09 Integro-partial differential equations
35R11 Fractional partial differential equations
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