An efficient numerical approach to solve a class of variable‐order fractional integro‐partial differential equations

2020 ◽  
Vol 37 (1) ◽  
pp. 674-689
Author(s):  
Afshin Babaei ◽  
Seddigheh Banihashemi ◽  
Carlo Cattani
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Approach ◽  
Variable Order ◽  
Partial Differential

scholarly journals An efficient numerical approach for space fractional partial differential equations

2020 ◽  
Vol 59 (5) ◽  
pp. 2911-2919
Author(s):  
Rabia Shikrani ◽  
M.S. Hashmi ◽  
Nargis Khan ◽  
Abdul Ghaffar ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Approach ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

scholarly journals New method for solving a class of fractional partial differential equations with applications

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 277-286 ◽  
Author(s):  
Hossein Jafari ◽  
Haleh Tajadodi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Linear Algebraic Equations ◽  
Spline Polynomial

In this work we suggest a numerical approach based on the B-spline polynomial to obtain the solution of linear fractional partial differential equations. We find the operational matrix for fractional integration and then we convert the main problem into a system of linear algebraic equations by using this matrix. Examples are provided to show the simplicity of our method.

scholarly journals Topics in structure-preserving discretization

Acta Numerica ◽  
2011 ◽  
Vol 20 ◽  
pp. 1-119 ◽  
Author(s):  
Snorre H. Christiansen ◽  
Hans Z. Munthe-Kaas ◽  
Brynjulf Owren
Keyword(s):  
Partial Differential Equations ◽  
Finite Elements ◽  
Differential Equations ◽  
Lie Group ◽  
Variable Order ◽  
Partial Differential ◽  
Structure Preserving ◽  
Hodge Laplacian ◽  
Lie Group Integrators ◽  
Inverse Systems

In the last few decades the concepts of structure-preserving discretization, geometric integration and compatible discretizations have emerged as subfields in the numerical approximation of ordinary and partial differential equations. The article discusses certain selected topics within these areas; discretization techniques both in space and time are considered. Lie group integrators are discussed with particular focus on the application to partial differential equations, followed by a discussion of how time integrators can be designed to preserve first integrals in the differential equation using discrete gradients and discrete variational derivatives.Lie group integrators depend crucially on fast and structure-preserving algorithms for computing matrix exponentials. Preservation of domain symmetries is of particular interest in the application of Lie group integrators to PDEs. The equivariance of linear operators and Fourier transforms on non-commutative groups is used to construct fast structure-preserving algorithms for computing exponentials. The theory of Weyl groups is employed in the construction of high-order spectral element discretizations, based on multivariate Chebyshev polynomials on triangles, simplexes and simplicial complexes.The theory of mixed finite elements is developed in terms of special inverse systems of complexes of differential forms, where the inclusion of cells corresponds to pullback of forms. The theory covers, for instance, composite piecewise polynomial finite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose product is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge–Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to variational discretizations.

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