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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.

The RC Circuit Described by Local Fractional Differential Equations

Zhao X.-H., Zhang Y., Zhao D., Yang X.

Fundamenta Informaticae

2017

Vol. 151, nr 1/4

419--429

A non-differentiable resistor-capacitor circuit comprised of the capacitor and resistor in the fractal-time domain is first proposed in this article. The solution behavior of the corresponding local fractional ordinary differential equation is presented for the Mittag-Leffler decay defined on Cantor sets. The obtained results reveal the sufficiency of the local fractional calculus in the analysis of the fractal electrical systems.

A New Family of the Local Fractional PDEs

Yang X.-J., Machado J. A. T., Nieto J. J.

Fundamenta Informaticae

2017

Vol. 151, nr 1/4

63--75

A new family of the local fractional PDEs is investigated in this article. The linear, quasilinear, semilinear and nonlinear local fractional PDEs are presented. Furthermore, three types of the local fractional PDEs are discussed, namely, parabolic, hyperbolic and elliptic. Several examples illustrate the corresponding models in nonlinear mathematical physics.