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.

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 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.

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.

scholarly journals Existence of a Unique Weak Solution to a Nonlinear Non-Autonomous Time-Fractional Wave Equation (of Distributed-Order)

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1283
Author(s):  
Karel Van Bockstal
Keyword(s):  
Wave Equation ◽  
Weak Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Order Differential Operator ◽  
Unique Weak Solution ◽  
Fractional Wave Equation ◽  
Equation Of Time ◽  
Initial Boundary Value ◽  
Distributed Order

We study an initial-boundary value problem for a fractional wave equation of time distributed-order with a nonlinear source term. The coefficients of the second order differential operator are dependent on the spatial and time variables. We show the existence of a unique weak solution to the problem under low regularity assumptions on the data, which includes weakly singular solutions in the class of admissible problems. A similar result holds true for the fractional wave equation with Caputo fractional derivative.

scholarly journals Error Analysis of An Explicit Finite Difference Approximation for the Space Fractional Wave Equations

2012 ◽  
Vol 17 (2) ◽  
pp. 245
Author(s):  
N.H. Sweilam ◽  
T.A. Assiri
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Error Analysis ◽  
Wave Equations ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Fractional Wave Equation ◽  
Fractional Wave Equations ◽  
Explicit Finite Difference ◽  
The Stability

In this paper, the space fractional wave equation (SFWE) is numerically studied, where the fractional derivative is defined in the sense of Caputo. An explicit finite difference approximation (EFDA) for SFWE is presented. The stability and the error analysis of the EFDA are discussed. To demonstrate the effectiveness of the approximated method, some test examples are presented.   

scholarly journals A linear finite difference scheme for the generalized dissipative symetric regularized long wave equation with damping

2019 ◽  
Vol 23 (Suppl. 3) ◽  
pp. 719-726 ◽  
Author(s):  
Xi Wang ◽  
Jin-Song Hu ◽  
Hong Zhang
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Initial Boundary Value

In this paper, we study and analyze a three-level linear finite difference scheme for the initial boundary value problem of the symmetric regularized long wave equation with damping. The proposed scheme has the second accuracy both for the spatial and temporal discretization. The convergence and stability of the numerical solutions are proved by the mathematical induction and the discrete functional analysis. Numerical results are given to verify the accuracy and the efficiency of proposed algorithm.

scholarly journals PRACTICAL ERROR ANALYSIS FOR THE THREE-LEVEL BILINEAR FEM AND FINITE-DIFFERENCE SCHEME FOR THE 1D WAVE EQUATION WITH NON-SMOOTH DATA

2018 ◽  
Vol 23 (3) ◽  
pp. 359-378
Author(s):  
Alexander Zlotnik ◽  
Olga Kireeva
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Error Analysis ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Free Term ◽  
The Rich ◽  
Initial Boundary Value

We deal with the standard three-level bilinear FEM and finite-difference scheme with a weight to solve the initial-boundary value problem for the 1D wave equation. We consider the rich collection of initial data and the free term which are the Dirac δ-functions, discontinuous, continuous but with discontinuous derivatives and from the Sobolev spaces, accomplish the practical error analysis in the L2, L1, energy and uniform norms as the mesh re_nes and compare results with known theoretical error bounds.

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