Weak solvability of the variable-order subdiffusion equation

2020 ◽  
Vol 23 (3) ◽  
pp. 920-934
Author(s):  
Andrii Hulianytskyi
Keyword(s):  
Time Derivative ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Order ◽  
Linear Partial Differential Equations ◽  
New Type ◽  
Subdiffusion Equation ◽  
Initial Boundary Value ◽  
The Laplace Operator ◽  
Weak Solvability

AbstractIn this work, we study a new type of linear partial differential equations – the variable-order subdiffusion equation. Here the Laplace operator in space acts on the Riemann-Liouville time derivative of space-dependent order. We construct a variable-order Sobolev and prove the weak solvability of the initial-boundary value problem for this equation, which confirms the well-posedness of the problem. Finally, we briefly discuss the application of the developed approach to the more general variable-order reaction-subdiffusion equation.

scholarly journals Numerical treatment for the solution of singularly perturbed pseudo-parabolic problem on an equidistributed grid

2020 ◽  
Vol 9 (1) ◽  
pp. 169-174 ◽  
Author(s):  
Jugal Mohapatra ◽  
Deepti Shakti
Keyword(s):  
Time Derivative ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Perturbation Parameter ◽  
Euler Method ◽  
Uniform Mesh ◽  
Central Difference ◽  
Monitor Function ◽  
The Time Domain ◽  
Initial Boundary Value

AbstractThe initial-boundary value problem for a pseudo-parabolic equation exhibiting initial layer is considered. For solving this problem numerically independence of the perturbation parameter, we propose a difference scheme which consists of the implicit-Euler method for the time derivative and a central difference method for the spatial derivative on uniform mesh. The time domain is discretized with a nonuniform grid generated by equidistributing a positive monitor function. The performance of the numerical scheme is tested which confirms the expected behavior of the method. The existing method is compared with other methods available in the recent literature.

scholarly journals Initial-boundary value problem for a time-fractional subdiffusion equation on the torus

2021 ◽  
Vol 65 (3) ◽  
pp. 17-24
Author(s):  
Ravshan Ashurov ◽  
◽  
Oqila Muhiddinova
Keyword(s):  
Boundary Value Problem ◽  
Initial Function ◽  
Boundary Value ◽  
Fourier Method ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Subdiffusion Equation ◽  
Initial Boundary Value ◽  
The Right

An initial-boundary value problem for a time-fractional subdiffusion equation with the Riemann-Liouville derivatives on N-dimensional torus is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. It should be noted, that the condition on the initial function found in this paper is less restrictive than the analogous condition in the case of an equation with derivatives in the sense of Caputo.

Error Analysis of a Finite Difference Method on Graded Meshes for a Multiterm Time-Fractional Initial-Boundary Value Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 815-825 ◽  
Author(s):  
Chaobao Huang ◽  
Xiaohui Liu ◽  
Xiangyun Meng ◽  
Martin Stynes
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Value Problem ◽  
Difference Method ◽  
Boundary Value ◽  
Time Derivative ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Graded Meshes ◽  
Initial Boundary Value

AbstractAn initial-boundary value problem, whose differential equation contains a sum of fractional time derivatives with orders between 0 and 1, is considered. Its spatial domain is {(0,1)^{d}} for some {d\in\{1,2,3\}}. This problem is a generalisation of the problem considered by Stynes, O’Riordan and Gracia in SIAM J. Numer. Anal. 55 (2017), pp. 1057–1079, where {d=1} and only one fractional time derivative was present. A priori bounds on the derivatives of the unknown solution are derived. A finite difference method, using the well-known L1 scheme for the discretisation of each temporal fractional derivative and classical finite differences for the spatial discretisation, is constructed on a mesh that is uniform in space and arbitrarily graded in time. Stability and consistency of the method and a sharp convergence result are proved; hence it is clear how to choose the temporal mesh grading in a optimal way. Numerical results supporting our theoretical results are provided.

scholarly journals Factorized difference scheme for two-dimensional subdiffusion equation in nonhomogeneous media

2016 ◽  
Vol 99 (113) ◽  
pp. 1-13 ◽  
Author(s):  
Aleksandra Delic ◽  
Sandra Hodzic ◽  
Bosko Jovanovic
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Nonhomogeneous Media ◽  
Subdiffusion Equation ◽  
Initial Boundary Value

A factorized finite-difference scheme for numerical approximation of initial-boundary value problem for two-dimensional subdiffusion equation in nonhomogeneous media is proposed. Its stability and convergence are investigated. The corresponding error bounds are obtained.

scholarly journals On Temporal Behaviour of Solutions in Thermoelasticity of Porous Micropolar Bodies

2014 ◽  
Vol 22 (1) ◽  
pp. 169-188 ◽  
Author(s):  
Marin Marin ◽  
Olivia Florea
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Time Derivative ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cesàro Means ◽  
Temporal Behaviour ◽  
Cesáro Means ◽  
Thermoelastic Body ◽  
Initial Boundary Value

AbstractWe consider a porous thermoelastic body, including voidage time derivative among the independent constitutive variables. For the initial boundary value problem of such materials, we analyze the temporal behaviour of the solutions. To this aim we use the Cesaro means for the components of energy and prove the asymptotic equipartition in mean of the kinetic and strain energies.

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