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.

scholarly journals A General Numerical Method for Analyzing the Linear Stability of Stratified Parallel Shear Flows

2014 ◽  
Vol 31 (12) ◽  
pp. 2795-2808 ◽  
Author(s):  
Tim Rees ◽  
Adam Monahan
Keyword(s):  
Stability Analysis ◽  
Numerical Method ◽  
Shear Flows ◽  
Numerical Solutions ◽  
Analytical Approximations ◽  
Onset Of Turbulence ◽  
Parallel Shear Flows ◽  
Critical Layers ◽  
The Stability ◽  
Stability Results

Abstract The stability analysis of stratified parallel shear flows is fundamental to investigations of the onset of turbulence in atmospheric and oceanic datasets. The stability analysis is performed by considering the behavior of small-amplitude waves, which is governed by the Taylor–Goldstein (TG) equation. The TG equation is a singular second-order eigenvalue problem, whose solutions, for all but the simplest background stratification and shear profiles, must be computed numerically. Accurate numerical solutions require that particular care be taken in the vicinity of critical layers resulting from the singular nature of the equation. Here a numerical method is presented for finding unstable modes of the TG equation, which calculates eigenvalues by combining numerical solutions with analytical approximations across critical layers. The accuracy of this method is assessed by comparison to the small number of stratification and shear profiles for which analytical solutions exist. New stability results from perturbations to some of these profiles are also obtained.

scholarly journals Numerical study of the Benjamin-Bona-Mahony-Burgers equation

2017 ◽  
Vol 35 (1) ◽  
pp. 127 ◽  
Author(s):  
M. Zarebnia
Keyword(s):  
Stability Analysis ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Numerical Study ◽  
Numerical Results ◽  
Spline Collocation ◽  
Unconditionally Stable ◽  
B Spline ◽  
Analysis Technique

In this paper, the quadratic B-spline collocation methodis implemented to find numerical solution of theBenjamin-Bona-Mahony-Burgers (BBMB) equation. Applying theVon-Neumann stability analysis technique, we show that the method is unconditionally stable. Also the convergence of the method is proved. The method is applied on some testexamples, and numerical results have been compared with theexact solution. The numerical solutions show theefficiency of the method computationally.

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.

The Solution of Lateral Heat Loss Problem Using Collocation Method with Cubic B-Splines Finite Element

2011 ◽  
Vol 110-116 ◽  
pp. 3184-3190
Author(s):  
Necdet Bildik ◽  
Duygu Dönmez Demir
Keyword(s):  
Finite Element ◽  
Stability Analysis ◽  
Heat Loss ◽  
Collocation Method ◽  
Numerical Solutions ◽  
Analytic Solutions ◽  
B Splines ◽  
Stability Method ◽  
The Stability ◽  
Lateral Heat

This paper deals with the solutions of lateral heat loss equation by using collocation method with cubic B-splines finite elements. The stability analysis of this method is investigated by considering Fourier stability method. The comparison of the numerical solutions obtained by using this method with the analytic solutions is given by the tables and the figure.

A Parallel Block Predictor-Corrector Method by Python-Based Distributed Computing

2012 ◽  
Vol 263-266 ◽  
pp. 1315-1318
Author(s):  
Kun Ming Yu ◽  
Ming Gong Lee
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Method ◽  
Initial Value Problem ◽  
Message Passing ◽  
Message Passing Interface ◽  
Parallel Structure ◽  
Initial Value ◽  
Speed Up ◽  
Predictor Corrector

This paper is to discuss how Python can be used in designing a cluster parallel computation environment in numerical solution of some block predictor-corrector method for ordinary differential equations. In the parallel process, MPI-2(message passing interface) is used as a standard of MPICH2 to communicate between CPUs. The operation of data receiving and sending are operated and controlled by mpi4py which is based on Python. Implementation of a block predictor-corrector numerical method with one and two CPUs respectively is used to test the performance of some initial value problem. Minor speed up is obtained due to small size problems and few CPUs used in the scheme, though the establishment of this scheme by Python is valuable due to very few research has been carried in this kind of parallel structure under Python.

scholarly journals Numerical Solutions of Fractional Integrodifferential Equations of Bratu Type by Using CAS Wavelets

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Mingxu Yi ◽  
Kangwen Sun ◽  
Jun Huang ◽  
Lifeng Wang
Keyword(s):  
Numerical Method ◽  
Error Estimation ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Integrodifferential Equations ◽  
Algebraic Equations ◽  
Approximation Solution ◽  
Cas Wavelets

A numerical method based on the CAS wavelets is presented for the fractional integrodifferential equations of Bratu type. The CAS wavelets operational matrix of fractional order integration is derived. A truncated CAS wavelets series together with this operational matrix is utilized to reduce the fractional integrodifferential equations to a system of algebraic equations. The solution of this system gives the approximation solution for the truncated limited2k(2M+1). The convergence and error estimation of CAS wavelets are also given. Two examples are included to demonstrate the validity and applicability of the approach.

scholarly journals Unconditional Stability of a Numerical Method for the Dual-Phase-Lag Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
M. A. Castro ◽  
J. A. Martín ◽  
F. Rodríguez
Keyword(s):  
Numerical Method ◽  
Numerical Solutions ◽  
Unconditional Stability ◽  
Bioheat Transfer ◽  
Phase Lag ◽  
Dual Phase ◽  
Stability Properties ◽  
The Stability ◽  
Transfer Problems ◽  
Uniformly Bounded

The stability properties of a numerical method for the dual-phase-lag (DPL) equation are analyzed. The DPL equation has been increasingly used to model micro- and nanoscale heat conduction in engineering and bioheat transfer problems. A discretization method for the DPL equation that could yield efficient numerical solutions of 3D problems has been previously proposed, but its stability properties were only suggested by numerical experiments. In this work, the amplification matrix of the method is analyzed, and it is shown that its powers are uniformly bounded. As a result, the unconditional stability of the method is established.

scholarly journals NUMERICAL SOLUTION OF TUMOR-IMMUNE INTERACTION USING 2-POINT BLOCK BACKWARD DIFFERENTIATION METHOD

2012 ◽  
Vol 09 ◽  
pp. 278-284 ◽  
Author(s):  
NOR AIN AZEANY MOHD NASIR ◽  
ZARINA BIBI IBRAHIM ◽  
MOHAMED SULEIMAN ◽  
K. I. OTHMAN ◽  
YONG FAEZAH RAHIM
Keyword(s):  
Numerical Solution ◽  
Numerical Solutions ◽  
Interaction Model ◽  
Numerical Results ◽  
Model Systems ◽  
Computational Time ◽  
Differentiation Formula ◽  
Backward Differentiation Formula ◽  
Differentiation Method ◽  
Interaction System

In this paper, we consider tumor-immune interaction model systems. The numerical solutions for the tumor-immune interaction system are obtained by using the 2-point Block Backward Differentiation Formula (BBDF) methods developed by Zarina et al. in 2007. The numerical results are presented in terms of computational time and accuracy of the solutions.

scholarly journals Analysis of a New Fractional Model for Damped Bergers’ Equation

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 35-41 ◽  
Author(s):  
Jagdev Singh ◽  
Devendra Kumar ◽  
Maysaa Al Qurashi ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Fractional Derivative ◽  
Numerical Solution ◽  
Numerical Results ◽  
Fractional Model ◽  
Applied Method ◽  
The Stability

AbstractIn this article, we present a fractional model of the damped Bergers’ equation associated with the Caputo-Fabrizio fractional derivative. The numerical solution is derived by using the concept of an iterative method. The stability of the applied method is proved by employing the postulate of fixed point. To demonstrate the effectiveness of the used fractional derivative and the iterative method, numerical results are given for distinct values of the order of the fractional derivative.

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