Asymptotic Solution of a Nonlinear Initial–Boundary Value Problem for the Diffusion Equation with a Small Parameter Multiplying the Time Derivative

2013 ◽  
Vol 25 (1) ◽  
pp. 9-26 ◽  
Author(s):  
A. A. Belolipetskii ◽  
E. A. Malinina
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Asymptotic Solution ◽  
Small Parameter ◽  
Boundary Value ◽  
Time Derivative ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value

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 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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