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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 $\left(1+2\right)$ and $\left(1+3$) 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.
##### MSC:
 35A35 Theoretical approximation to solutions of PDE 45D05 Volterra integral equations 65R20 Integral equations (numerical methods) 35R09 Integro-partial differential equations 35R11 Fractional partial differential equations
##### Keywords:
Galerkin method; anomalous diffusion