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 A Second Order Weighted Numerical Scheme on Nonuniform Meshes for Convection Diffusion Parabolic Problems

2020 ◽  
Author(s):  
Lolugu Govindarao ◽  
Jugal Mohapatra
Keyword(s):  
Time Derivative ◽  
Perturbation Parameter ◽  
Rectangular Domain ◽  
Shishkin Mesh ◽  
Second Order ◽  
Parabolic Problems ◽  
Uniform Mesh ◽  
Central Difference ◽  
Convection Diffusion ◽  
Convection Diffusion Equation

In this article, a singularly perturbed parabolic convection-diffusion equation on a rectangular domain is considered. The solution of the problem possesses regular boundary layer which appears in the spatial variable. To discretize the time derivative, we use two type of schemes, first the implicit Euler scheme and second the implicit trapezoidal scheme on a uniform mesh. For approximating the spatial derivatives, we use the monotone hybrid scheme, which is a combination of midpoint upwind scheme and central difference scheme with variable weights on Shishkin-type meshes (standard Shishkin mesh, Bakhvalov-Shishkin mesh and modified Bakhvalov-Shishkin mesh). We prove that both numerical schemes converge uniformly with respect to the perturbation parameter and are of second order accurate. Thomas algorithm is used to solve the tri-diagonal system. Finally, to support the theoretical results, we present a numerical experiment by using the proposed methods.

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.

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.

Convergence in Positive Time for a Finite Difference Method Applied to a Fractional Convection-Diffusion Problem

2018 ◽  
Vol 18 (1) ◽  
pp. 33-42 ◽  
Author(s):  
José Luis Gracia ◽  
Eugene O’Riordan ◽  
Martin Stynes
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Diffusion Problem ◽  
Initial Time ◽  
Uniform Mesh ◽  
Convection Diffusion ◽  
Initial Boundary Value

AbstractA standard finite difference method on a uniform mesh is used to solve a time-fractional convection-diffusion initial-boundary value problem. Such problems typically exhibit a mild singularity at the initial time {t=0}. It is proved that the rate of convergence of the maximum nodal error on any subdomain that is bounded away from {t=0} is higher than the rate obtained when the maximum nodal error is measured over the entire space-time domain. Numerical results are provided to illustrate the theoretical error bounds.

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