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Invariance of integro-differential equations with moving region of integration

100%

Zawistowski Z. J.

tom Vol. 47, no 2

103-112

General criterion of invariance of integro-differential equations under the Lie symmetry group of point transformations is derived. It is a generalization of the previous form of the criterion to the case of a moving range of integration. This is the situation when a region of integration depends on external, with respect to integration, variables what leads to its explicit dependence on a group parameter, so the region of integration moves under symmetry transformations. General case of dependence on independent and dependent variables and their derivatives is considered.

The implicit numerical method for the one-dimensional anomalous subdiffusion equation with a nonlinear source term

84%

Błasik M.

tom Vol. 69, nr 6

art. no. e138240

In the paper, the numerical method of solving the one-dimensional subdiffusion equation with the source term is presented. In the approach used, the key role is played by transforming of the partial differential equation into an equivalent integro-differential equation. As a result of the discretization of the integro-differential equation obtained an implicit numerical scheme which is the generalized Crank-Nicolson method. The implicit numerical schemes based on the finite difference method, such as the Carnk-Nicolson method or the Laasonen method, as a rule are unconditionally stable, which is their undoubted advantage. The discretization of the integro-differential equation is performed in two stages. First, the left-sided Riemann-Liouville integrals are approximated in such a way that the integrands are linear functions between successive grid nodes with respect to the time variable. This allows us to find the discrete values of the integral kernel of the left-sided Riemann-Liouville integral and assign them to the appropriate nodes. In the second step, second order derivative with respect to the spatial variable is approximated by the difference quotient. The obtained numerical scheme is verified on three examples for which closed analytical solutions are known.

Existence results of noninstantaneous impulsive fractional integro-differential equation

84%

Kataria H. R. , Patel P. H. , Shah V.

2020

tom Vol. 53, nr 1

373--384

Existence of mild solution for noninstantaneous impulsive fractional order integro-differential equations with local and nonlocal conditions in Banach space is established in this paper. Existence results with local and nonlocal conditions are obtained through operator semigroup theory using generalized Banach contraction theorem and Krasnoselskii’s fixed point theorem, respectively. Finally, illustrations are added to validate derived results.