scholarly journals Numerical Solution of Partial Integrodifferential Equations of Diffusion Type

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Imran Aziz ◽  
Imran Khan
Keyword(s):  
Applied Sciences ◽  
Dynamical Systems ◽  
Numerical Solution ◽  
Numerical Method ◽  
Collocation Method ◽  
Numerical Results ◽  
Integrodifferential Equations ◽  
Benchmark Problems ◽  
Diffusion Type ◽  
One Dimensional

A collocation method based on linear Legendre multiwavelets is developed for numerical solution of one-dimensional parabolic partial integrodifferential equations of diffusion type. Such equations have numerous applications in many problems in the applied sciences to model dynamical systems. The proposed numerical method is validated by applying it to various benchmark problems from the existing literature. The numerical results confirm the accuracy, efficiency, and robustness of the proposed method.

scholarly journals A Collocation Method for the Numerical Solution of Nonlinear Fractional Dynamical Systems

Algorithms ◽  
2019 ◽  
Vol 12 (8) ◽  
pp. 156 ◽  
Author(s):  
Francesca Pitolli
Keyword(s):  
Applied Sciences ◽  
Dynamical Systems ◽  
Numerical Methods ◽  
Numerical Solution ◽  
Collocation Method ◽  
Fractional Order ◽  
Time Derivative ◽  
Test Problems ◽  
Fractional Differential

Fractional differential problems are widely used in applied sciences. For this reason, there is a great interest in the construction of efficient numerical methods to approximate their solution. The aim of this paper is to describe in detail a collocation method suitable to approximate the solution of dynamical systems with time derivative of fractional order. We will highlight all the steps necessary to implement the corresponding algorithm and we will use it to solve some test problems. Two Mathematica Notebooks that can be used to solve these test problems are provided.

scholarly journals Solving Volterra Integrodifferential Equations via Diagonally Implicit Multistep Block Method

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Nur Auni Baharum ◽  
Zanariah Abdul Majid ◽  
Norazak Senu
Keyword(s):  
Stability Analysis ◽  
Numerical Solution ◽  
Numerical Method ◽  
Numerical Computation ◽  
Numerical Solutions ◽  
Numerical Results ◽  
Integrodifferential Equations ◽  
Block Method ◽  
The Stability ◽  
Predictor Corrector

The performance of the numerical computation based on the diagonally implicit multistep block method for solving Volterra integrodifferential equations (VIDE) of the second kind has been analyzed. The numerical solutions of VIDE will be computed at two points concurrently using the proposed numerical method and executed in the predictor-corrector (PECE) mode. The strategy to obtain the numerical solution of an integral part is discussed and the stability analysis of the diagonally implicit multistep block method was investigated. Numerical results showed the competence of diagonally implicit multistep block method when solving Volterra integrodifferential equations compared to the existing methods.

Numerical Solution of Diffusion and Reaction–Diffusion Partial Integro-Differential Equations

2018 ◽  
Vol 15 (06) ◽  
pp. 1850047 ◽  
Author(s):  
Imran Aziz ◽  
Imran Khan
Keyword(s):  
Population Dynamics ◽  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Method ◽  
Reaction Diffusion ◽  
Nonlinear Problems ◽  
Haar Wavelet ◽  
Benchmark Problems ◽  
Two Dimensional ◽  
One Dimensional

In this paper, a collocation method based on Haar wavelet is developed for numerical solution of diffusion and reaction–diffusion partial integro-differential equations. The equations are parabolic partial integro-differential equations and we consider both one-dimensional and two-dimensional cases. Such equations have applications in several practical problems including population dynamics. An important advantage of the proposed method is that it can be applied to both linear as well as nonlinear problems with slide modification. The proposed numerical method is validated by applying it to various benchmark problems from the existing literature. The numerical results confirm the accuracy, efficiency and robustness of the proposed method.

An Upwind Numerical Method for a Hyperbolic One-Dimensional Two-Fluid Model

2011 ◽  
Author(s):  
Youn-Gyu Jung ◽  
Moon-Sun Chung ◽  
Sung-Jae Yi
Keyword(s):  
Numerical Method ◽  
Fluid Model ◽  
Equation System ◽  
Benchmark Problems ◽  
Two Phase ◽  
One Dimensional ◽  
Flow Problems ◽  
Complete Set ◽  
Two Fluid Model ◽  
Two Fluid

This study discusses on the implementation of an upwind method for a one-dimensional two-fluid model including the surface tension effect in the momentum equations. This model consists of a complete set of six equations including two-mass, two-momentum, and two-internal energy conservation equations having all real eigenvalues. Based on this equation system with upwind numerical method, the present authors first make a pilot code and then solve some benchmark problems to verify whether this model and numerical method is able to properly solve some fundamental one-dimensional two-phase flow problems or not.

scholarly journals Taylor collocation method for a system of nonlinear Volterra delay integro-differential equations with application to COVID-19 epidemic

2020 ◽  
Author(s):  
Hafida Laib ◽  
Azzeddine Bellour ◽  
Aissa Boulmerka
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Method ◽  
Collocation Method ◽  
Mathematical Ecology ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Taylor Polynomials ◽  
Convergent Algorithm ◽  
A Current

Abstract The present paper deals with the numerical solution for a general form of a system of nonlinear Volterra delay integro-differential equations (VDIDEs). The main purpose of this work is to provide a current numerical method based on the use of continuous collocation Taylor polynomials for the numerical solution of nonlinear VDIDEs systems. It is shown that this method is convergent. Numerical results will be presented to prove the validity and effectiveness of this convergent algorithm. We apply two models to the COVID-19 epidemic in China and one for the Predator-Prey model in mathematical ecology.

scholarly journals Computer modeling system for the numerical solution of the one-dimensional non-stationary Burgers’ equation

2019 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Numerical Solution ◽  
Computer Modeling ◽  
Burgers Equation ◽  
Boundary Value ◽  
Three Dimensional ◽  
Benchmark Problems ◽  
One Dimensional ◽  
Radial Basis ◽  
The One

The computer modeling system for numerical solution of the nonlinear one-dimensional non-stationary Burgers’ equation is described. The numerical solution of the Burgers’ equation is obtained by a meshless scheme using the method of partial solutions and radial basis functions. Time discretization of the one-dimensional Burgers’ equation is obtained by the generalized trapezoidal method (θ-scheme). The inverse multiquadric function is used as radial basis functions in the computer modeling system. The computer modeling system allows setting the initial conditions and boundary conditions as well as setting the source function as a coordinate- and time-dependent function for solving partial differential equation. A computer modeling system allows setting such parameters as the domain of the boundary-value problem, number of interpolation nodes, the time interval of non-stationary boundary-value problem, the time step size, the shape parameter of the radial basis function, and coefficients in the Burgers’ equation. The solution of the nonlinear one-dimensional non-stationary Burgers’ equation is visualized as a three-dimensional surface plot in the computer modeling system. The computer modeling system allows visualizing the solution of the boundary-value problem at chosen time steps as three-dimensional plots. The computational effectiveness of the computer modeling system is demonstrated by solving two benchmark problems. For solved benchmark problems, the average relative error, the average absolute error, and the maximum error have been calculated.

Unsteady Loads on Supercavitating Hydrofoils of Finite Span

1966 ◽  
Vol 10 (02) ◽  
pp. 107-118
Author(s):  
Sheila Evans Widnall
Keyword(s):  
Numerical Solution ◽  
Numerical Method ◽  
Integral Equations ◽  
Digital Computer ◽  
High Speed ◽  
Three Dimensional ◽  
Numerical Results ◽  
Oscillatory Motion ◽  
Surface Theory ◽  
Finite Span

Linearized three-dimensional lifting-surface theory is derived for a supercavitating hydrofoil with finite span in steady or oscillatory motion through an infinite fluid. The resulting coupled-integral equations are solved on a high-speed digital computer using a numerical method of assumed modes similar to that used for fully wetted surfaces. Numerical results for lift and moment for both steady and oscillating foils are compared with other theories and experiments. Results of these calculations indicate that this numerical solution gives an efficient and accurate prediction of loads on a supercavitating foil.

scholarly journals Solving Hyperchaotic Systems Using the Spectral Relaxation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
S. S. Motsa ◽  
P. G. Dlamini ◽  
M. Khumalo
Keyword(s):  
Dynamical Systems ◽  
Numerical Solution ◽  
Numerical Method ◽  
Pseudospectral Method ◽  
Relaxation Method ◽  
Chebyshev Pseudospectral Method ◽  
Spectral Relaxation Method ◽  
Spectral Relaxation ◽  
Chebyshev Pseudospectral ◽  
Hyperchaotic Systems

A new multistage numerical method based on blending a Gauss-Siedel relaxation method and Chebyshev pseudospectral method, for solving complex dynamical systems exhibiting hyperchaotic behavior, is presented. The proposed method, called the multistage spectral relaxation method (MSRM), is applied for the numerical solution of three hyperchaotic systems, namely, the Chua, Chen, and Rabinovich-Fabrikant systems. To demonstrate the performance of the method, results are presented in tables and diagrams and compared to results obtained using a Runge-Kutta-(4,5)-based MATLAB solver,ode45, and other previously published results.

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