Analytical solutions for multi-term time-space coupling fractional delay partial differential equations with mixed boundary conditions

This paper studies the problem -Δu + λu = up in nonsmooth domains with mixed boundary conditions. Special attention will be given here to the critical case and to domains which have no further regularity than a Lipschitzian boundary. For such domains, we obtain a generalized version of Cherrier's inequality and prove an existence result. This was achieved by using an extended definition of the manifold.

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Solving partial differential equations in deformed grids by estimating local average gradients with planes

Journal of Physics Conference Series ◽

10.1088/1742-6596/2090/1/012069 ◽

2021 ◽

Vol 2090 (1) ◽

pp. 012069

Author(s):

Aarne Pohjonen

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Boundary Conditions ◽

Analytical Solutions ◽

Physical Science ◽

Partial Differential ◽

Average Gradient ◽

Initial And Boundary Conditions ◽

Local Average

Abstract For constructing physical science based models in irregular numerical grids, an easy-to-implement method for solving partial differential equations has been developed and its accuracy has been evaluated by comparison to analytical solutions that are available for simple initial and boundary conditions. The method is based on approximating the local average gradients of a field by fitting equation of plane to the field quantities at neighbouring grid positions and then calculating an estimate for the local average gradient from the plane equations. The results, comparison to analytical solutions, and accuracy are presented for 2-dimensional cases.

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Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2018-0019 ◽

2018 ◽

Vol 21 (2) ◽

pp. 312-335 ◽

Cited By ~ 10

Author(s):

Xiao-Li Ding ◽

Juan J. Nieto

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Fractional Differential Equations ◽

Analytical Solutions ◽

Laplacian Operator ◽

Fractional Partial Differential Equations ◽

Partial Differential ◽

Time Space ◽

Fractional Differential ◽

Time Fractional Differential Equations

AbstractIn this paper, we consider the analytical solutions of multi-term time-space fractional partial differential equations with nonlocal damping terms for general mixed Robin boundary conditions on a finite domain. Firstly, method of reduction to integral equations is used to obtain the analytical solutions of multi-term time fractional differential equations with integral terms. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the multi-term time-space fractional partial differential equations with nonlocal damping terms to the multi-term time fractional differential equations with integral terms. By applying the obtained analytical solutions to the resulting multi-term time fractional differential equations with integral terms, the desired analytical solutions of the multi-term time-space fractional partial differential equations with nonlocal damping terms are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.

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