Partial fractional derivatives of Riesz type and nonlinear fractional differential equations

Tarasov, Vasily E.

doi:10.1007/s11071-016-2991-y

Cited by 20 publications

(7 citation statements)

References 87 publications

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“…[1][2][3][4][5][6][7][8] Recent examples are the heat transport in fractal media, 9 fractal hydrodynamic equations, 10 fractal electrostatics, 11 fractal FokkerPlanck equations 12 and fractal description of stress and strain in elasticity. 13 There are several alternative approaches for describing the complex and fractal behaviors in nature.…”

Section: Introductionmentioning

confidence: 99%

“…13 There are several alternative approaches for describing the complex and fractal behaviors in nature. [1][2][3][4][5][6][7][8] The theory of the local fractional derivative (LFD) is a mathematical tool for describing fractals, that was used to model the fractal complexity in shallow water surfaces, 14 LC-electric circuit, 15 traveling-wave solution of the Burgerstype equation, 16 PDEs, [17][18][19][20] ODEs, 21 and inequalities. 22,23 The useful models for the LFD were considered [24][25][26][27][28][29] and discussed.…”

Section: Introductionmentioning

confidence: 99%

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Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain

2017

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The new Boussinesq-type model in a fractal domain is derived based on the formulation of the local fractional derivative. The novel traveling wave transform of the non-differentiable type is adopted to convert the local fractional Boussinesq equation into a nonlinear local fractional ODE. The exact traveling wave solution is also obtained with aid of the non-differentiable ¶ Corresponding author. This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited. X. -J. Yang, J. A. T. Machado & D. Baleanu graph. The proposed method, involving the fractal special functions, is efficient for finding the exact solutions of the nonlinear PDEs in fractal domains. 1740006-1

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain

2017

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“…In recent years, the fractional differential operators and equations have increasingly attracted much attention, since they are good at describing memory and heredity of some complex systems compared with the integer-order derivative [1,2]. So far, the fractional differential operators have been applied in various research fields, such as optimization [3], fractional quantum mechanics [4], finance [5], image process [6], and biomedical engineering [7].…”

Section: Introductionmentioning

confidence: 99%

Existence and Uniqueness of Weak Solutions to Variable-Order Fractional Laplacian Equations with Variable Exponents

Guo

2021

Journal of Function Spaces

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In this paper, the variable-order fractional Laplacian equations with variable exponents and the Kirchhoff-type problem driven by p · -fractional Laplace with variable exponents were studied. By using variational method, the authors obtain the existence and uniqueness results.

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“…Fractional calculus which is an ancient field and equally important like calculus of integer order is a branch of mathematical analysis that studies the properties of defining real number powers or complex number powers of the differentiation operator [12,38]. In comparison with the integer-order derivatives, fractional derivative can describe different complex dynamical systems more accurately, since it simultaneously possesses memory, which makes it a powerful tool in modeling physical phenomena related to nonlocality and memory effect [2,33,51,56].…”

Section: Introductionmentioning

confidence: 99%

Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

Wei

Zhai

Zhang

2020

Commun. Appl. Math. Comput.

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The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems. The scheme is discretized by a weighted-shifted Grünwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction. The stability and the L 2-convergence of the scheme are proved for all variable-order (t) ∈ (0, 1). The proposed method is of accuracyorder O(3 + h k+1) , where , h, and k are the temporal step size, the spatial step size, and the degree of piecewise P k polynomials, respectively. Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.

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Partial fractional derivatives of Riesz type and nonlinear fractional differential equations

Cited by 20 publications

References 87 publications

Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain

Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain

Existence and Uniqueness of Weak Solutions to Variable-Order Fractional Laplacian Equations with Variable Exponents

Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

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