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Bandrowski, Bartosz; Karczewska, Anna; Rozmej, Piotr

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

Int. J. Appl. Math. Comput. Sci. 20, No. 2, 261-269 (2010).
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.
Reviewer: Titus Petrila (Cluj-Napoca)

Cited in 8 Documents

MSC:

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

Keywords:

Galerkin method; anomalous diffusion
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\textit{B. Bandrowski} et al., Int. J. Appl. Math. Comput. Sci. 20, No. 2, 261--269 (2010; Zbl 1201.35020)
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References:

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