scholarly journals Finite difference approximation of a parabolic problem with variable coefficients

2014 ◽  
Vol 95 (109) ◽  
pp. 49-62 ◽  
Author(s):  
Bosko Jovanovic ◽  
Zorica Milovanovic
Keyword(s):  
Finite Difference ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Variable Coefficients ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Convergence Rate Estimate

We study the convergence of a finite difference scheme that approximates the third initial-boundary-value problem for a parabolic equation with variable coefficients on a unit square. We assume that the generalized solution of the problem belongs to the Sobolev space W s,s/2 2, s?3. An almost second-order convergence rate estimate (with additional logarithmic factor) in the discrete W 1,1/2 2 norm is obtained. The result is based on some nonstandard a priori estimates involving fractional order discrete Sobolev norms.

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.

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.

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.

A diffusion–convection problem with a fractional derivative along the trajectory of motion

2021 ◽  
Vol 36 (3) ◽  
pp. 157-163
Author(s):  
Alexander V. Lapin ◽  
Vladimir V. Shaidurov
Keyword(s):  
Finite Difference ◽  
Fractional Derivative ◽  
Characteristic Curve ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Finite Difference Schemes ◽  
Stability Estimates ◽  
Initial Boundary Value

Abstract A new mathematical model of the diffusion–convective process with ‘memory along the flow path’ is proposed. This process is described by a homogeneous one-dimensional Dirichlet initial-boundary value problem with a fractional derivative along the characteristic curve of the convection operator. A finite-difference approximation of the problem is constructed and investigated. The stability estimates for finite-difference schemes are proved. The accuracy estimates are given for the case of sufficiently smooth input data and the solution.

scholarly journals Finite difference approximation for parabolic interface problem with time-dependent coefficients

2016 ◽  
Vol 99 (113) ◽  
pp. 67-76
Author(s):  
Bratislav Sredojevic ◽  
Dejan Bojovic
Keyword(s):  
Finite Difference ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sobolev Norm ◽  
Time Dependent ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Interface Problem ◽  
Two Dimensional ◽  
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 W~12,1/2 Sobolev norm, compatible with the smoothness of the coefficients and solution, is proved.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

scholarly journals On initial boundary value problem with Dirichlet integral conditions for a hyperbolic equation with the Bessel operator

2003 ◽  
Vol 2003 (10) ◽  
pp. 487-502
Author(s):  
Abdelfatah Bouziani
Keyword(s):  
Hyperbolic Equation ◽  
A Priori ◽  
Weighted Sobolev Spaces ◽  
A Priori Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Bessel Operator ◽  
Integral Conditions ◽  
Initial Boundary Value

We consider a mixed problem with Dirichlet and integral conditions for a second-order hyperbolic equation with the Bessel operator. The existence, uniqueness, and continuous dependence of a strongly generalized solution are proved. The proof is based on an a priori estimate established in weighted Sobolev spaces and on the density of the range of the operator corresponding to the abstract formulation of the considered problem.

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