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

Abstract

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.

Normalized finite fractional differences: Computational and accuracy breakthroughs

“…That promising solution has, however, suffered from the necessity to select dominant OBF pole(s), which may significantly affect the approximation accuracy. It is finally worth mentioning about yet another, numerical approach to the approximation of fractional systems in the context of integral equations (Momani and Odibat, 2007;Bandrowski et al, 2010;Saeedi et al, 2011), the approach 908 R. Stanisławski and K.J. Latawiec suffering, again, from a computational burden.…”

Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type

“…If A(t) is a nonsingular matrix for all t ∈ I, then multiplying (1) by A −1 changes it to a system of Volterra integral equations of the second kind, whose theoretical and numerical analysis has been already investigated (see, e.g., Atkinson, 2001;Hochstadt, 1973;Bandrowski et al, 2010;Saeedi et al, 2011). If A(t) is a singular matrix with constant rank for all t ∈ I, then the system (1) will be an IAE or a singular system of Volterra integral equations of the fourth kind, and if A(t) is a singular matrix with constant rank for some t ∈ I, then the system (1) will be a singular system of Volterra integral equations of the third kind or weakly singular Volterra integral equations.…”

On Fractional Schrödinger Equation

“…Recently, working on some other problems, we have developed a numerical method for solving integral equations of convolution type [13]. Such integral equations appeared to be (under some assumptions concerning boundary conditions) equivalent to differential evolution equations with fractional time but with a fractional derivative of the order α ∈ [1,2].…”