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Diffusion equation including a local fractional derivative and weighted inner product

Cetinkaya Suleyman, Demir Ali

Journal of Applied Mathematics and Computational Mechanics

2022

Vol. 21, nr 1

19--27

In this research, we discuss the construction of the analytic solution of the homogenous initial boundary value problem including partial differential equations of fractional order. Since the homogenous initial boundary value problem involves a local fractional order derivative, it has classical initial and boundary conditions. By means of separation of the variables method and the inner product defined on L2 [0, l], the solution is constructed in the form of a Fourier series including the exponential function. The illustrative examples present the applicability and influence of the separation of variables method on time fractional diffusion problems. Moreover, as the fractional order α tends to 1, the solution of the fractional diffusion problem tends to the solution of the diffusion problem which proves the accuracy of the solution.

Solution of hybrid time fractional diffusion problem via weighted inner product

Cetinkaya Suleyman, Demir Ali

Journal of Applied Mathematics and Computational Mechanics

2021

Vol. 20, nr 2

17--27

In this research, we discuss the construction of the analytic solution of homogenous initial boundary value problem including partial differential equations of fractional order. Since the homogenous initial boundary value problem involves the Hybrid fractional order derivative with various coefficients functions, it has classical initial and boundary conditions. By means of separation of the variables method and the inner product defined on L 2 [0,l], the solution is constructed in the form of a Fourier series including the bivariate Mittag-Leffler function. An illustrative example presents the applicability and influence of the separation of variables method on time fractional diffusion problems. Moreover, as the fractional order α tends to 1, the solution of the fractional diffusion problem tends to the solution of the diffusion problem which proves the accuracy of the solution.

Fourth boundary condition for electromagnetic field

Spałek D.

Journal of Technical Physics

2000

Vol. 41, no 2

129-144

The paper deals with the problem of boundary condition formulation for an electromagnetic field. The nonlinear boundary condition has been proposed. This condition has been introduced on the basis of Maxwell's equations. The nonlinear condition includes three well-known boundary conditions of Dirichlet, Neumann and Hankel. An example has been provided to show the usefulness of the boundary condition proposed. The fourth boundary condition enables us to reduce the region of numerical analysis.