Numerical and analytical investigation for solutions of fractional Oskolkov–Benjamin–Bona–Mahony–Burgers equation describing propagation of long surface waves

2021 ◽  
Author(s):  
B Sagar ◽  
S. Saha Ray
Keyword(s):  
Finite Difference ◽  
Numerical Scheme ◽  
Soliton Solutions ◽  
Type Equation ◽  
Analytical Investigation ◽  
Stability And Convergence ◽  
Kudryashov Method ◽  
Collocation Approach ◽  
The Stability ◽  
Kansa Method

In this paper, a novel meshless numerical scheme to solve the time-fractional Oskolkov–Benjamin–Bona–Mahony–Burgers-type equation has been proposed. The proposed numerical scheme is based on finite difference and Kansa-radial basis function collocation approach. First, the finite difference scheme has been employed to discretize the time-fractional derivative and subsequently, the Kansa method is utilized to discretize the spatial derivatives. The stability and convergence analysis of the time-discretized numerical scheme are also elucidated in this paper. Moreover, the Kudryashov method has been utilized to acquire the soliton solutions for comparison with the numerical results. Finally, numerical simulations are performed to confirm the applicability and accuracy of the proposed scheme.

Numerical soliton solutions of fractional (2 + 1)-dimensional Nizhnik–Novikov–Veselov equations in nonlinear optics

2021 ◽  
Vol 35 (06) ◽  
pp. 2150090
Author(s):  
B Sagar ◽  
S. Saha Ray
Keyword(s):  
Finite Difference ◽  
Radial Basis Function ◽  
Basis Function ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Soliton Solutions ◽  
Numerical Examples ◽  
Kudryashov Method ◽  
Tanh Method ◽  
Kansa Method

In this paper, time-fractional (2 + 1)-dimensional Nizhnik–Novikov–Veselov equations have been solved numerically utilizing the Kansa method, in which the multiquadrics are taken as radial basis function. To attain this, a numerical scheme based on finite difference and Kansa method has been proposed. In addition, the soliton solutions have been obtained by employing Kudryashov method and tanh method for comparison purpose with the obtained numerical solutions. The numerical examples are given to demonstrate the accuracy and applicability of the proposed method.

scholarly journals Stability and Convergence of Two-Step Obrechkoff Scheme For Second-Order Two-Point Boundary Value Problem

2017 ◽  
Vol 4 (1) ◽  
Author(s):  
Olufemi Bosede ◽  
Ashiribo Wusu ◽  
Moses Akanbi
Keyword(s):  
Mathematical Modeling ◽  
Differential Equation ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Finite Difference Methods ◽  
Stability And Convergence ◽  
Point Boundary ◽  
Difference Methods ◽  
The Stability

Mathematical modeling of scientific and engineering processes often yield Boundary Value Problems (BVPs). One of the broad categories of numerical methods for solving BVPs is the finite difference methods, in which the differential equation is replaced by a set of difference equations which are solved by direct or iterative methods. In this paper, we use some properties of matrices to analyze the stability and convergence of the prominent finite difference methods - two-step Obrechkoff method - for solving the boundary value problem $u^{\prime \prime} = f(t,u)$, $a < x < b$, $u(a) = \eta_1$, $u(b) = \eta_2$. Conditions for the stability and convergence of the two-step Obrechkoff method method were established.

scholarly journals A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel

2021 ◽  
Vol 5 (3) ◽  
pp. 85
Author(s):  
Tayyaba Akram ◽  
Zeeshan Ali ◽  
Faranak Rabiei ◽  
Kamal Shah ◽  
Poom Kumam
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Numerical Study ◽  
Stability And Convergence ◽  
Weakly Singular Kernel ◽  
Integro Differential Equation ◽  
Suggested Technique ◽  
Collocation Approach ◽  
The Stability ◽  
Efficiency And Reliability

Fractional differential equations can present the physical pathways with the storage and inherited properties due to the memory factor of fractional order. The purpose of this work is to interpret the collocation approach for tackling the fractional partial integro-differential equation (FPIDE) by employing the extended cubic B-spline (ECBS). To determine the time approximation, we utilize the Caputo approach. The stability and convergence analysis have also been analyzed. The efficiency and reliability of the suggested technique are demonstrated by two numerical applications, which support the theoretical results and the effectiveness of the implemented algorithm.

scholarly journals Explicit Finite-Difference Scheme for the Numerical Solution of the Model Equation of Nonlinear Hereditary Oscillator with Variable-Order Fractional Derivatives

2016 ◽  
Vol 26 (3) ◽  
pp. 429-435 ◽  
Author(s):  
Roman I. Parovik
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Fractional Derivatives ◽  
Stability And Convergence ◽  
Variable Order ◽  
The Difference ◽  
Explicit Finite Difference ◽  
The Stability ◽  
Variable Order Fractional Derivatives

Abstract The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is stable and converges to the exact solution.

The Finite Difference Scheme for 3d Mathematical Modeling of a Wood Drying Process

2001 ◽  
Vol 1 (2) ◽  
pp. 125-137 ◽  
Author(s):  
Raimondas Čiegis ◽  
Vadimas Starikovičius
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Experiments ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Drying Process ◽  
The Stability ◽  
Robin Boundary

AbstractThis work discusses issues on the design and analysis of finite difference schemes for 3D modeling the process of moisture motion in the wood. A new finite difference scheme is proposed. The stability and convergence in the maximum norm are proved for Robin boundary conditions. The influence of boundary conditions is investigated, and results of numerical experiments are presented.

Numerical soliton solutions of fractional modied (2 + 1)-dimensional Konopelchenko-Dubrovsky equations in plasma physics

2021 ◽  
Author(s):  
Santanu Saha Ray ◽  
B Sagar
Keyword(s):  
Solitary Wave ◽  
Plasma Physics ◽  
Exact Solutions ◽  
Radial Basis Function ◽  
Basis Function ◽  
Numerical Scheme ◽  
Soliton Solutions ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Kansa Method

Abstract In this paper, the time-fractional modied (2+1)-dimensional Konopelchenko-Dubrovsky equations have been solved numerically using the Kansa method, in which the multiquadrics used as radial basis function. To achieve this, a numerical scheme based on nite dierenceand Kansa method has been proposed. Also the solitary wave solutions have been obtained by using Kudryashov technique. The computed results are compared with the exact solutions as well as with the soliton solutions obtained by Kudryashov technique to show the accuracy of the proposed method.

scholarly journals A Dufort-Frankel Difference Scheme for Two-Dimensional Sine-Gordon Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22 ◽  
Author(s):  
Zongqi Liang ◽  
Yubin Yan ◽  
Guorong Cai
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
The Stability ◽  
Predictor Corrector ◽  
Theoretical Results ◽  
Methods Numerical

A standard Crank-Nicolson finite-difference scheme and a Dufort-Frankel finite-difference scheme are introduced to solve two-dimensional damped and undamped sine-Gordon equations. The stability and convergence of the numerical methods are considered. To avoid solving the nonlinear system, the predictor-corrector techniques are applied in the numerical methods. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

scholarly journals Some Remarks on the Stability Condition of Numerical Scheme of the KdV-type Equation

2017 ◽  
Vol 9 (4) ◽  
pp. 11 ◽  
Author(s):  
Chun-Te Lee ◽  
Jeng-Eng Lin ◽  
Chun-Che Lee ◽  
Mei-Li Liu
Keyword(s):  
Stability Condition ◽  
Numerical Scheme ◽  
Kdv Equation ◽  
Type Equation ◽  
Stability Criteria ◽  
Numerical Schemes ◽  
The Kdv Equation ◽  
The Stability ◽  
Difference Fourier ◽  
Central Finite Difference

This paper has employed a comparative study between the numerical scheme and stability condition. Numerical calculations are carried out based on three different numerical schemes, namely the central finite difference, fourier leap-frog, and fourier spectral RK4 schemes. Stability criteria for different numerical schemes are developed for the KdV equation, and numerical examples are put to test to illustrate the accuracy and stability between the solution profile and numerical scheme.

A preconditioned fast finite difference method for space-time fractional partial differential equations

2017 ◽  
Vol 20 (1) ◽  
Author(s):  
Hongfei fu ◽  
Hong Wang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Scheme ◽  
Mathematical Structure ◽  
Space Time ◽  
Optimal Order ◽  
Conservation Property ◽  
Long Time ◽  
The Stability

AbstractWe develop a fast space-time finite difference method for space-time fractional diffusion equations by fully utilizing the mathematical structure of the scheme. A circulant block preconditioner is proposed to further reduce the computational costs. The method has optimal-order memory requirement and approximately linear computational complexity. The method is not lossy, as no compression of the underlying numerical scheme has been employed. Consequently, the method retains the stability, accuracy, and, in particular, the conservation property of the underlying numerical scheme. Numerical experiments are presented to show the efficiency and capacity of long time modelling of the new method.

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