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Numerical solution of Burgers' equation with modified cubic B-spline differential quadrature method

APPLIED MATHEMATICS AND COMPUTATION 2013年 224卷 166-177页

作者： Arora, Geeta Singh, Brajesh Kumar Graph Era Hill Univ Sch Allied Sci Dept Math Dehra Dun 248002 Uttarakhand India

In this paper, a new numerical method, "modified cubic-B-spline differential quadrature method (MCB-DQM)" is proposed to find the approximate solution of the Burgers' equation. The modified cubic-B-spline basis functions are used in differential quadrature to determine the weighting coefficients. The MCB-DQM is used in space, and the optimal four-stage, order three strong stability-preserving time-stepping Runge-Kutta (SSP-RK43) scheme is used in time for solving the resulting system of ordinary differential equations. To check the efficiency and accuracy of the method, four examples of Burgers' equation are included with their numerical solutions, L-2 and L-infinity errors and comparisons are done with the results given in the literature. The proposed method produces better results as compared to the results obtained by almost all the schemes available in the literature, and approaching to the exact solutions. The presented method is seen to be easy, powerful, efficient and economical to implement as compared to the existing techniques for finding the numerical solutions for various kinds of linear/nonlinear physical models. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Burgers' equation Cubic B-spline Modified cubic B-spline Differential quadrature method (DQM) thomas algorithm SSP-RK43 scheme

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A Computational Analysis to Burgers Huxley Equation

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Computers, Materials & Continua 2021年第5期67卷 2161-2183页

作者： Muhammad Saqib Muhammad Shoaib Arif Shahid Hasnain Daoud S.Mashat Department of Mathematics Numl UniversityIslamabad44000Pakistan Department of Mathematics Air UniversityIslamabad44000Pakistan Department of Mathematics Air UniversityMultan CampusMultan66000Pakistan Department of Mathematics King Abdulaziz UniversityJeddahSaudi Arabia

The efficiency of solving computationally partial differential equations can be profoundly highlighted by the creation of precise,higher-order compact numerical scheme that results in truly outstanding accuracy at a given *** objective of this article is to develop a highly accurate novel algorithm for two dimensional non-linear Burgers Huxley(BH)*** proposed compact numerical scheme is found to be free of superiors approximate oscillations across discontinuities,and in a smooth ow region,it efciently obtained a high-order *** particular,two classes of higherorder compact nite difference schemes are taken into account and compared based on their computational *** stability and accuracy show that the schemes are unconditionally stable and accurate up to a two-order in time and to six-order in ***,algorithms and data tables illustrate the scheme efciency and decisiveness for solving such non-linear coupled *** is scaled in terms of L_(2) and L_(∞) norms,which validate the approximated results with the corresponding analytical *** investigation of the stability requirements of the implicit method applied in the algorithm was carried *** agreement was constructed under indistinguishable computational *** proposed methods can be implemented for real-world problems,originating in engineering and science.

关键词： Burgers Huxley equation nite difference schemes HOC schemes thomas algorithm Von-Neumann stability ana

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The Preissmann box scheme and its modification for transcritical flows

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2007年第7期70卷 791-811页

作者： Freitag, M. A. Morton, K. W. Univ Bath Dept Math Sci Bath BA2 7AY Avon England Univ Oxford Comp Lab Oxford OX1 3QD England

The Preissmann box scheme is the standard numerical method used by hydraulic engineers to model open channel flows or surcharged flows in pipes;but it breaks down for transcritical flows. Various alternatives have been suggested for such cases, but we show how simple modifications of the difference scheme and its method of solution enable both to be used for such flows. Analysis of the double-sweeping thomas algorithm shows both how it works in the subcritical case and the effectiveness of its modification for transcritical flows. A well-known analytical transcritical flow is used to demonstrate the accuracy of the new numerical scheme. Copyright (c) 2006 John Wiley & Sons, Ltd.

关键词： box scheme transcritical flow St Venant equations shallow water flow thomas algorithm Newton method

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A numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations

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ENGINEERING COMPUTATIONS 2017年第4期34卷 1257-1276页

作者： Alshomrani, Ali Saleh Pandit, Sapna Alzahrani, Abdullah K. Alghamdi, Metib Said Jiwari, Ram King Abdulaziz Univ Fac Sci Dept Math Jeddah Saudi Arabia Indian Inst Technol Roorkee Dept Math Roorkee Uttar Pradesh India Jazan Univ Fac Sci Dept Math Jazan Saudi Arabia

Purpose - The main purpose of this work is the development of a numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations. These types of equations describe a variety of physical models in the vibrations of structures, nonlinear optics, quantum field theory and solid-state physics, etc. Design/methodology/approach - Dirichlet boundary conditions cannot be handled easily by cubic trigonometric B-spline functions. Then, a modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and a numerical algorithm is developed. The proposed algorithm reduced the hyperbolic-type wave equations into a system of first-order ordinary differential equations (ODEs) in time variable. Then, stability-preserving SSP-RK54 scheme and the thomas algorithm are used to solve the obtained system. The stability of the algorithm is also discussed. Findings - A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from the schemes developed (Abbas et al., 2014;Nazir et al., 2016) and depicts the computational modelling of hyperbolic-type wave equations. Originality/value - To the best of the authors' knowledge, this technique is novel for solving hyperbolic-type wave equations and the developed algorithm is free from quasi-linearization process and finite difference operators for time derivatives. This algorithm gives better results than the results discussed in literature (Dehghan and Shokri, 2008;Batiha et al., 2007;Mittal and Bhatia, 2013;Jiwari, 2015).

关键词： Stability Cubic trigonometric B-splines basis functions Hyperbolic-type wave equations SSP-RK54 scheme thomas algorithm Modified cubic trigonometric B-splines basis functions

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Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method

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APPLIED MATHEMATICS AND COMPUTATION 2013年 220卷 496-506页

作者： Mittal, R. C. Bhatia, Rachna IIT Roorkee Dept Math Roorkee 247667 Uttarakhand India

The present paper uses new approach and methodology to solve second order one dimensional hyperbolic telegraph equation numerically by B-spline collocation method. It is based on collocation of modified cubic B-spline basis functions over the finite elements. The given equation is decomposed into system of equations and modified cubic B-spline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of differential equations. The resulting system of equation subsequently has been solved by SSP-RK54 scheme. The efficacy of proposed approach has been confirmed with numerical experiments, which shows the results obtained are acceptable and in good agreement with earlier studies. The advantage of this scheme is that it can be conveniently used to solve the complex problems and it is also capable of reducing the size of computational work. (c) 2013 Elsevier Inc. All rights reserved.

关键词： Telegraph equation Modified cubic-Bspline basis function thomas algorithm SSP-RK54 scheme

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A new algorithm based on modified trigonometric cubic B-splines functions for nonlinear Burgers'-type equations

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INTERNATIONAL JOURNAL OF NUMERICAL METHODS FOR HEAT & FLUID FLOW 2017年第8期27卷 1638-1661页

作者： Jiwari, Ram Alshomrani, Ali Saleh Indian Inst Technol Roorkee Dept Math Roorkee Uttar Pradesh India King Abdulaziz Univ Fac Sci Dept Math Jeddah Saudi Arabia

Purpose - The main aim of the paper is to develop a new B-splines collocation algorithm based on modified cubic trigonometric B-spline functions to find approximate solutions of nonlinear parabolic Burgers'-type equations with Dirichlet boundary conditions. Design/methodology/approach - A modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and an algorithm is developed with the help of modified cubic trigonometric B-spline functions. The proposed algorithm reduced the Burgers' equations into a system of first-order nonlinear ordinary differential equations in time variable. Then, strong stability preserving RungeKutta 3rd order (SSP-RK3) scheme is used to solve the obtained system. Findings - A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from to the schemes developed in Abbas et al. (2014) and Nazir et al. (2016), and the developed algorithms are free from linearization process and finite difference operators. Originality/value - To the best knowledge of the authors, this technique is novel for solving nonlinear partial differential equations, and the new proposed technique gives better results than the results discussed in Ozis et al. (2003), Kutluay et al. (1999), Khater et al. (2008), Korkmaz and Dag (2011), Kutluay et al. (2004), Rashidi et al. (2009), Mittal and Jain (2012), Mittal and Jiwari (2012), Mittal and Tripathi (2014), Xie et al. (2008) andKadalbajoo et al. (2005).

关键词： Cubic trigonometric B-splines basis functions Modified cubic trigonometric B-splines basis functions Burgers' equation SSP-RK3 scheme thomas algorithm

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B-splines methods with redefined basis functions for solving fourth order parabolic partial differential equations

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APPLIED MATHEMATICS AND COMPUTATION 2011年第23期217卷 9741-9755页

作者： Mittal, R. C. Jain, R. K. IIT Dept Math Roorkee 247667 Uttarakhand India

In this work, we discuss two methods for solving a fourth order parabolic partial differential equation. In Method-I, we decompose the given equation into a system of second order equations and solve them by using cubic B-spline method with redefined basis functions. In Method-II, the equation is solved directly by applying quintic B-spline method with redefined basis functions. Stability of these methods have been discussed. Both methods are unconditionally stable. These methods are tested on four examples. The computed results are compared wherever possible with those already available in literature. We have developed Method-I for fourth order non homogeneous parabolic partial differential equation from which we can obtain displacement and bending moment both simultaneously, while Method-II gives only displacement. The results show that the derived methods are easily implemented and approximate the exact solution very well. (C) 2011 Elsevier Inc. All rights reserved.

关键词： Fourth order parabolic partial differential equation Cubic B-spline method with redefined basis functions Quintic B-spline method with redefined basis functions thomas algorithm

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Numerical solution of the coupled viscous Burgers' equation

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2011年第3期16卷 1304-1313页

作者： Mittal, R. C. Arora, Geeta Indian Inst Technol Roorkee Dept Math Roorkee 247667 Uttar Pradesh India

In the present paper, a numerical method is proposed for the numerical solution of a coupled system of viscous Burgers' equation with appropriate initial and boundary conditions, by using the cubic B-spline collocation scheme on the uniform mesh points. The scheme is based on Crank-Nicolson formulation for time integration and cubic B-spline functions for space integration. The method is shown to be unconditionally stable using von-Neumann method. The accuracy of the proposed method is demonstrated by applying it on three test problems. Computed results are depicted graphically and are compared with those already available in the literature. The obtained numerical solutions indicate that the method is reliable and yields results compatible with the exact solutions. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Coupled viscous Burgers' equation Collocation Cubic B-spline Bi-tridiagonal thomas algorithm

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Numerical Study to Coupled Three Dimensional Reaction Diffusion System

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IEEE ACCESS 2019年 7卷 46695-46705页

作者： Hasnain, Shahid Saqib, Muhammad Afzaal, Muhammad F. Hussain, Iqtadar PIEAS DPAM Nilore 45650 Pakistan Air Univ Dept Math Islamabad 44230 Pakistan Xi An Jiao Tong Univ Sch Chem Engn & Technol Xian 710049 Shaanxi Peoples R China Qatar Univ Dept Math Stat & Phys Doha 2713 Qatar

The numerical solution of reaction diffusion systems may require more computational efforts if the change in concentrations occurs extremely rapid. This is because more time points are needed to resolve the reaction diffusion process accurately. In this paper, three finite difference implicit schemes are used which are unconditionally stable in order to enhance consistency. Novelty is reported by compact finite difference implicit scheme on a reaction diffusion system with higher accuracy measured by L-2, L-infinity, and Relative(error) norms. Efficiency is observed by reducing grid space along small temporal steps. CPU performance, transmission capacity along comparison of three schemes shows excellent agreement with the analytical solution.

关键词： Reaction diffusion systems alternating direction implicit Douglas scheme higher order compact scheme thomas algorithm

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Numerical solution of nonlinear system of Klein-Gordon equations by cubic B-spline collocation method

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2015年第10期92卷 2139-2159页

作者： Mittal, R. C. Bhatia, Rachna IIT Roorkee Dept Math Roorkee 247667 Uttarakhand India

A technique to approximate the solutions of nonlinear Klein-Gordon equation and Klein-Gordon-Schrodinger equations is presented separately. The approach is based on collocation of cubic B-spline functions. The above-mentioned equations are decomposed into a system of partial differential equations, which are further converted to an amenable system of ODEs. The obtained system has been solved by SSP-RK54 scheme. Numerical solutions are presented for five examples, to show the accuracy and usefulness of proposed approach. The approximate solutions of both the equations are computed without using any transformation and linearization. The technique can be applied with ease to solve linear and nonlinear PDEs and also reduces the computational work.

关键词： Klein-Gordon equation coupled Klein-Gordon-Schrodinger equations thomas algorithm modified cubic B-spline basis function SSP-RK54 scheme

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