scholarly journals On Unconditionally Stable New Modified Fractional Group Iterative Scheme for the Solution of 2D Time-Fractional Telegraph Model

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2078
Author(s):  
Ajmal Ali ◽  
Thabet Abdeljawad ◽  
Azhar Iqbal ◽  
Tayyaba Akram ◽  
Muhammad Abbas
Keyword(s):  
Dirichlet Boundary ◽  
Iterative Scheme ◽  
Relaxation Method ◽  
Approximate Solutions ◽  
Dirichlet Boundary Conditions ◽  
Group Method ◽  
Numerical Tests ◽  
The Matrix ◽  
Fractional Group ◽  
The Stability

In this study, a new modified group iterative scheme for solving the two-dimensional (2D) fractional hyperbolic telegraph differential equation with Dirichlet boundary conditions is obtained from the 2h-spaced standard and rotated Crank–Nicolson FD approximations. The findings of new four-point modified explicit group relaxation method demonstrates the rapid rate of convergence of proposed method as compared to the existing schemes. Numerical tests are performed to test the capability of the group iterative scheme in comparison with the point iterative scheme counterparts. The stability of the derived modified group method is proven by the matrix norm algorithm. The obtained results are tabulated and concluded that exact solutions are exactly symmetric with approximate solutions.

FACTORIZATION OF THE CONVECTION-DIFFUSION OPERATOR AND THE SCHWARZ ALGORITHM

1995 ◽  
Vol 05 (01) ◽  
pp. 67-93 ◽  
Author(s):  
F. NATAF ◽  
F. ROGIER
Keyword(s):  
Boundary Conditions ◽  
Rate Of Convergence ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Interface Conditions ◽  
Diffusion Operator ◽  
Convection Diffusion ◽  
Numerical Tests ◽  
Schwarz Algorithm ◽  
Theoretical Results

In the original Schwarz algorithm, Dirichlet boundary conditions are used as interface conditions. We consider the use of the operators arising from the factorization of the convection-diffusion operator as transmission conditions. The rate of convergence is then significantly higher. Theoretical results are proven and numerical tests are shown.

scholarly journals Stability of Non-Linear Dirichlet Problems with ϕ-Laplacian

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 647
Author(s):  
Michał Bełdziński ◽  
Marek Galewski ◽  
Igor Kossowski
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Dirichlet Boundary Conditions ◽  
Laplacian Operator ◽  
Dirichlet Problems ◽  
Non Linear ◽  
The Stability

We study the stability and the solvability of a family of problems −(ϕ(x′))′=g(t,x,x′,u)+f* with Dirichlet boundary conditions, where ϕ, u, f* are allowed to vary as well. Applications for boundary value problems involving the p-Laplacian operator are highlighted.

scholarly journals Qualitative Analysis of Implicit Dirichlet Boundary Value Problem for Caputo-Fabrizio Fractional Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Rozi Gul ◽  
Muhammad Sarwar ◽  
Kamal Shah ◽  
Thabet Abdeljawad ◽  
Fahd Jarad
Keyword(s):  
Differential Equations ◽  
Qualitative Analysis ◽  
Fractional Differential Equations ◽  
Dirichlet Boundary ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Dirichlet Boundary Conditions ◽  
Fractional Differential ◽  
The Stability ◽  
Implicit Fractional Differential Equations

This article studies a class of implicit fractional differential equations involving a Caputo-Fabrizio fractional derivative under Dirichlet boundary conditions (DBCs). Using classical fixed-point theory techniques due to Banch’s and Krasnoselskii, a qualitative analysis of the concerned problem for the existence of solutions is established. Furthermore, some results about the stability of the Ulam type are also studied for the proposed problem. Some pertinent examples are given to justify the results.

scholarly journals HARMONIC FUNCTIONS AND THE SPECTRUM OF THE LAPLACIAN ON THE SIERPINSKI CARPET

Fractals ◽  
2013 ◽  
Vol 21 (01) ◽  
pp. 1350002 ◽  
Author(s):  
MATTHEW BEGUÉ ◽  
TRISTAN KALLONIATIS ◽  
ROBERT S. STRICHARTZ
Keyword(s):  
Boundary Conditions ◽  
Harmonic Functions ◽  
Dirichlet Boundary ◽  
Approximate Solutions ◽  
Dirichlet Boundary Conditions ◽  
Small Cells ◽  
Graph Structure ◽  
Boundary Values ◽  
Sierpinski Carpet ◽  
Sierpiński Carpet

Kusuoka and Zhou have defined the Laplacian on the Sierpinski carpet using average values of functions on small cells and the graph structure of cell adjacency. We have implemented an algorithm that uses their method to approximate solutions to boundary value problems. As a result we have a wealth of data concerning harmonic functions with prescribed boundary values, and eigenfunctions of the Laplacian with either Neumann or Dirichlet boundary conditions. We will present some of this data and discuss some ideas for defining normal derivatives on the boundary of the carpet.

scholarly journals An Improved Computationally Efficient Method for Finding the Drazin Inverse

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Haifa Bin Jebreen ◽  
Yurilev Chalco-Cano
Keyword(s):  
Efficient Method ◽  
Matrix Theory ◽  
Iterative Scheme ◽  
Drazin Inverse ◽  
Stability And Convergence ◽  
Computationally Efficient ◽  
Initial Matrix ◽  
The Matrix ◽  
The Stability

Drazin inverse is one of the most significant inverses in the matrix theory, where its computation is an intensive and useful task. The objective of this work is to propose a computationally effective iterative scheme for finding the Drazin inverse. The convergence is investigated analytically by applying a suitable initial matrix. The theoretical discussions are upheld by several experiments showing the stability and convergence of the proposed method.

scholarly journals Multigrid for Q k Finite Element Matrices Using a (Block) Toeplitz Symbol Approach

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 5 ◽  
Author(s):  
Paola Ferrari ◽  
Ryma Imene Rahla ◽  
Cristina Tablino-Possio ◽  
Skander Belhaj ◽  
Stefano Serra-Capizzano
Keyword(s):  
Dirichlet Boundary ◽  
Elliptic Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
One Dimension ◽  
Higher Dimension ◽  
Matrix Size ◽  
Polynomial Degree ◽  
Optimal Behavior ◽  
Higher Dimensional ◽  
The Matrix

In the present paper, we consider multigrid strategies for the resolution of linear systems arising from the Q k Finite Elements approximation of one- and higher-dimensional elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div − a ( x ) ∇ · , with a continuous and positive over Ω ¯ , Ω being an open and bounded subset of R 2 . While the analysis is performed in one dimension, the numerics are carried out also in higher dimension d ≥ 2 , showing an optimal behavior in terms of the dependency on the matrix size and a substantial robustness with respect to the dimensionality d and to the polynomial degree k.

scholarly journals An Improved Version of Residual Power Series Method for Space-Time Fractional Problems

2022 ◽  
Vol 2022 ◽  
pp. 1-9
Author(s):  
Mine Aylin Bayrak ◽  
Ali Demir ◽  
Ebru Ozbilge
Keyword(s):  
Power Series ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Space Time ◽  
Dirichlet Boundary Conditions ◽  
Power Series Method ◽  
Residual Power Series ◽  
Residual Power

The task of present research is to establish an enhanced version of residual power series (RPS) technique for the approximate solutions of linear and nonlinear space-time fractional problems with Dirichlet boundary conditions by introducing new parameter λ . The parameter λ allows us to establish the best numerical solutions for space-time fractional differential equations (STFDE). Since each problem has different Dirichlet boundary conditions, the best choice of the parameter λ depends on the problem. This is the major contribution of this research. The illustrated examples also show that the best approximate solutions of various problems are constructed for distinct values of parameter λ . Moreover, the efficiency and reliability of this technique are verified by the numerical examples.

On a Nonlinear System of Reaction-Diffusion Equations

2006 ◽  
Vol 11 (2) ◽  
pp. 115-121 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Weak Solution ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic System ◽  
Positive Parameter ◽  
Nonlinear Elliptic ◽  
Positive Weak Solution

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,− div(|∇u1|p1−2∇u1) = λu1α11u2α12... unα1n,   x ∈ Ω,− div(|∇u2|p2−2∇u2) = λu1α21u2α22... unα2n,   x ∈ Ω, ... , − div(|∇un|pn−2∇un) = λu1αn1u2αn2... unαnn,   x ∈ Ω,ui = 0,   x ∈ ∂Ω,   i = 1, 2, ..., n,  where λ is a positive parameter, Ω is a bounded domain in RN (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < pi < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.

scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

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