On the decomposition of solutions: From fractional diffusion to fractional Laplacian

2021 ◽  
Vol 24 (5) ◽  
pp. 1571-1600
Author(s):  
Yulong Li
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Ordinary Differential Equations ◽  
Laplace Equation ◽  
Fractional Derivatives ◽  
Fractional Laplacian ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Deep Insight ◽  
Structure Of Solutions

Abstract This paper investigates the structure of solutions to the BVP of a class of fractional ordinary differential equations involving both fractional derivatives (R-L or Caputo) and fractional Laplacian with variable coefficients. This family of equations generalize the usual fractional diffusion equation and fractional Laplace equation. We provide a deep insight to the structure of the solutions shared by this family of equations. The specific decomposition of the solution is obtained, which consists of the “good” part and the “bad” part that precisely control the regularity and singularity, respectively. Other associated properties of the solution will be characterized as well.

scholarly journals Numerical methods for solving the multi-term time-fractional wave-diffusion equation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Fawang Liu ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Pinghui Zhuang ◽  
Qingxia Liu
Keyword(s):  
Numerical Methods ◽  
Theoretical Analysis ◽  
Diffusion Equation ◽  
Fractional Derivatives ◽  
Fractional Laplacian ◽  
Numerical Results ◽  
Diffusion Equations ◽  
Time Space ◽  
Methods And Techniques ◽  
Wave Diffusion

AbstractIn this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

A Matrix-Vector Operation-Based Numerical Solution Method for Linear m-th Order Ordinary Differential Equations: Application to Engineering Problems

2013 ◽  
Vol 5 (03) ◽  
pp. 269-308 ◽  
Author(s):  
M. Aminbaghai ◽  
M. Dorn ◽  
J. Eberhardsteiner ◽  
B. Pichler
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Solution Method ◽  
Variable Coefficients ◽  
Exact Integration ◽  
Robust Solution ◽  
Vector Operation ◽  
Engineering Sciences ◽  
Engineering Problems ◽  
Matrix Vector

AbstractMany problems in engineering sciences can be described by linear, inhomogeneous,m-th order ordinary differential equations (ODEs) with variable coefficients. For this wide class of problems, we here present a new, simple, flexible, and robust solution method, based on piecewise exact integration of local approximation polynomials as well as on averaging local integrals. The method is designed for modern mathematical software providing efficient environments for numerical matrix-vector operation-based calculus. Based on cubic approximation polynomials, the presented method can be expected to perform (i) similar to the Runge-Kutta method, when applied to stiff initial value problems, and (ii) significantly better than the finite difference method, when applied to boundary value problems. Therefore, we use the presented method for the analysis of engineering problems including the oscillation of a modulated torsional spring pendulum, steady-state heat transfer through a cooling web, and the structural analysis of a slender tower based on second-order beam theory. Related convergence studies provide insight into the satisfying characteristics of the proposed solution scheme.

Hermite Pseudospectral Method for the Time Fractional Diffusion Equation with Variable Coefficients

2017 ◽  
Vol 18 (5) ◽  
pp. 385-393
Author(s):  
Zeting Liu ◽  
Shujuan Lü
Keyword(s):  
Diffusion Equation ◽  
Pseudospectral Method ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Fractional Diffusion Equation ◽  
Stability And Convergence ◽  
Initial Value ◽  
Fully Discrete ◽  
Time Fractional Diffusion Equation ◽  
Gauss Points

Abstract:We consider the initial value problem of the time fractional diffusion equation on the whole line and the fractional derivative is described in Caputo sense. A fully discrete Hermite pseudospectral approximation scheme is structured basing Hermite-Gauss points in space and finite difference in time. Unconditionally stability and convergence are proved. Numerical experiments are presented and the results conform to our theoretical analysis.

Elasticity Solution of Laminated Cylindrical Shell with Piezoelectric Layer under Local Ring Load

2007 ◽  
Vol 334-335 ◽  
pp. 917-920 ◽  
Author(s):  
M.H. Yas ◽  
Morteza Shakeri ◽  
M.R. Saviz
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Local Ring ◽  
Longitudinal Direction ◽  
Variable Coefficients ◽  
Elasticity Solution ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
External Forces

Elasticity solution is presented for simply-supported, orthotropic, piezoelectric cylindrical shell with finite length under local ring load in the middle of shell and electrostatic excitation. The highly coupled partial differential equations (p.d.e.) are reduced to ordinary differential equations(o.d.e.) with variable coefficients by means of trigonometric function expansion in longitudinal direction for displacement and external forces. The resulting ordinary differential equations are solved by Galerkin finite element method. Numerical examples are presented for [0/90/P] lamination with sensor and actuator for different thicknesses.

NONEXISTENCE OF SOLUTIONS RESULTS FOR CERTAIN FRACTIONAL DIFFERENTIAL EQUATIONS

2011 ◽  
Vol 16 (3) ◽  
pp. 488-497
Author(s):  
Mohamed Berbiche
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Fractional Differential Equations ◽  
Weak Solutions ◽  
Sufficient Conditions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Convective Term ◽  
Nonexistence Of Solutions ◽  
Fractional Differential

This paper is meant to establish sufficient conditions for the nonexistence of weak solutions to nonlinear fractional diffusion equation in space and time with nonlinear convective term. The Fujita exponent is determined.

scholarly journals Initial boundary value problems for a multi-term time fractional diffusion equation with generalized fractional derivatives in time

2021 ◽  
Vol 6 (11) ◽  
pp. 12114-12132
Author(s):  
Shuang-Shuang Zhou ◽  
◽  
Saima Rashid ◽  
Asia Rauf ◽  
Khadija Tul Kubra ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Spectral Problem ◽  
Direct Problem ◽  
Fractional Derivatives ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Time Fractional Diffusion Equation ◽  
Initial Boundary Value ◽  
The Given

<abstract><p>For a multi-term time-fractional diffusion equation comprising Hilfer fractional derivatives in time variables of different orders between $ 0 $ and $ 1 $, we have studied two problems (direct problem and inverse source problem). The spectral problem under consideration is self-adjoint. The solution to the given direct and inverse source problems is formulated utilizing the spectral problem. For the solution of the given direct problem, we proposed existence, uniqueness, and stability results. The existence, uniqueness, and consistency effects for the solution of the given inverse problem were addressed, as well as an inverse source for recovering space-dependent source term at certain $ T $. For the solution of the challenges, we proposed certain relevant cases.</p></abstract>

scholarly journals On the Mohand Transform and Ordinary Differential Equations with Variable Coefficients

2020 ◽  
Vol 35 (1) ◽  
pp. 01-06
Author(s):  
Mohamed E. Attaweel ◽  
Haneen Almassry
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Integral Equations ◽  
Integral Transform ◽  
Variable Coefficients ◽  
Sumudu Transforms ◽  
Differential And Integral Equations

The Mohand transform is a new integral transform introduced by Mohand M. Abdelrahim Mahgoub to facilitate the solution of differential and integral equations. In this article, a new integral transform, namely Mohand transform was applied to solve ordinary differential equations with variable coefficients by using the modified version of Laplace and Sumudu transforms.

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