The behavior of a finite difference approximation to a singular initial boundary value problem

2000 ◽  
Vol 76 (1-2) ◽  
pp. 115-130
Author(s):  
Azmy S. Ackleh ◽  
Lan Ke
Keyword(s):  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Initial Boundary Value

scholarly journals INVESTIGATION OF HYPERELASTIC SOFT SHELLS NONSTATIONARY DYNAMICS PROBLEMS SOLUTION FEATURES

2021 ◽  
Vol 83 (2) ◽  
pp. 151-159
Author(s):  
E.A. Korovaytseva
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Equations Of Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Difference Approximation ◽  
Analytical Studies ◽  
Initial Boundary Value

Results of hyperelastic soft shells nonlinear axisymmetric dynamic deforming problems solution algorithm testing are represented in the work. Equations of motion are given in vector-matrix form. For the nonlinear initial-boundary value problem solution an algorithm which lies in reduction of the system of partial differential equations of motion to the system of ordinary differential equations with the help of lines method is developed. At this finite-difference approximation of partial time derivatives is used. The system of ordinary differential equations obtained as a result of this approximation is solved using parameter differentiation method at each time step. The algorithm testing results are represented for the case of pressure uniformly distributed along the meridian of the shell and linearly increasing in time. Three types of elastic potential characterizing shell material are considered: Neo-hookean, Mooney – Rivlin and Yeoh. Features of numerical realization of the algorithm used are pointed out. These features are connected both with the properties of soft shells deforming equations system and with the features of the algorithm itself. The results are compared with analytical solution of the problem considered. Solution behavior at critical pressure value is investigated. Formulations and conclusions given in analytical studies of the problem are clarified. Applicability of the used algorithm to solution of the problems of soft shells dynamic deforming in the range of displacements several times greater than initial dimensions of the shell and deformations much greater than unity is shown. The numerical solution of the initial boundary value problem of nonstationary dynamic deformation of the soft shell is obtained without assumptions about the limitation of displacements and deformations. The results of the calculations are in good agreement with the results of analytical studies of the test problem.

Finite Difference Approximation of a Generalized Time-Fractional Telegraph Equation

2020 ◽  
Vol 20 (4) ◽  
pp. 595-607 ◽  
Author(s):  
Aleksandra Delić ◽  
Boško S. Jovanović ◽  
Sandra Živanović
Keyword(s):  
Finite Difference ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Telegraph Equation ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Fractional Telegraph Equation ◽  
Telegraph Equations ◽  
Initial Boundary Value ◽  
Generalized Time

AbstractWe consider a class of a generalized time-fractional telegraph equations. The existence of a weak solution of the corresponding initial-boundary value problem has been proved. A finite difference scheme approximating the problem is proposed, and its stability is proved. An estimate for the rate of convergence, in special discrete energetic Sobolev’s norm, is obtained. The theoretical results are confirmed by numerical examples.

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 Finite difference approximation of strong solutions of a parabolic interface problem on disconnected domains

2008 ◽  
Vol 84 (98) ◽  
pp. 37-48 ◽  
Author(s):  
Bosko Jovanovic ◽  
Lubin Vulkov
Keyword(s):  
Finite Difference ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Interface Problem ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value ◽  
Disconnected Domains

We investigate an initial boundary value problem for one dimensional parabolic equation in two disconnected intervals. A finite difference scheme for its solution is proposed and investigated. Convergence rate estimate compatible with the smoothness of input data is obtained.

scholarly journals Finite difference approximation for the 2D heat equation with concentrated capacity

Filomat ◽  
2018 ◽  
Vol 32 (20) ◽  
pp. 6979-6987
Author(s):  
Bratislav Sredojevic ◽  
Dejan Bojovic
Keyword(s):  
Finite Difference ◽  
Heat Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Dependent ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Two Dimensional ◽  
Sobolev Norms ◽  
Initial Boundary Value

The convergence of difference scheme for two-dimensional initial-boundary value problem for the heat equation with concentrated capacity and time-dependent coefficients of the space derivatives, is considered. An estimate of the rate of convergence in a special discrete Sobolev norms , compatible with the smoothness of the coefficients and solution, is proved.

Finite Difference Approximation of Fractional Wave Equation with Concentrated Capacity

2017 ◽  
Vol 17 (1) ◽  
pp. 33-49 ◽  
Author(s):  
Aleksandra Delić ◽  
Boško S. Jovanović
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Difference Approximation ◽  
Generalized Solutions ◽  
Difference Approximation ◽  
Dirac Delta ◽  
Fractional Wave Equation ◽  
Initial Boundary Value

AbstractWe consider the time fractional wave equation with coefficient which contains the Dirac delta distribution. The existence of generalized solutions of this initial-boundary value problem is proved. An implicit finite difference scheme approximating the problem is developed and its stability is proved. Estimates for the rate of convergence in special discrete energetic Sobolev norms are obtained. A numerical example confirms the theoretical results.

Sign in / Sign up

Export Citation Format

Share Document