Search results

1 – 10 of over 3000
Click here to view access options
Article
Publication date: 23 November 2018

Neeraj Dhiman and Mohammad Tamsir

The purpose of this paper is to present a new method, namely, “Re-modified quintic B-spline collocation method” to solve the Kuramoto–Sivashinsky (KS) type equations. In…

Abstract

Purpose

The purpose of this paper is to present a new method, namely, “Re-modified quintic B-spline collocation method” to solve the Kuramoto–Sivashinsky (KS) type equations. In this method, re-modified quintic B-spline functions and the Crank–Nicolson formulation is used for space and time integration, respectively. Five examples are considered to test out the efficiency and accuracy of the method. The main objective is to develop a method which gives more accurate results and reduces the computational cost so that the authors require less memory storage.

Design/methodology/approach

A new collocation technique is developed to solve the KS type equations. In this technique, quintic B-spline basis functions are re-modified and used to integrate the space derivatives while time derivative is discretized by using Crank–Nicolson formulation. The discretization yields systems of linear equations, which are solved by using Gauss elimination method with partial pivoting.

Findings

Five examples are considered to test out the efficiency and accuracy of the method. Finally, the present study summarizes the following outcomes: first, the computational cost of the proposed method is the less than quintic B-spline collocation method. Second, the present method produces better results than those obtained by Lattice Boltzmann method (Lai and Ma, 2009), quintic B-spline collocation method (Mittal and Arora, 2010), quintic B-spline differential quadrature method (DQM) (Mittal and Dahiya, 2017), extended modified cubic B-spline DQM (Tamsir et al., 2016) and modified cubic B-splines collocation method (Mittal and Jain, 2012).

Originality/value

The method presented in this paper is new to best of the authors’ knowledge. This work is the original work of authors and the manuscript is not submitted anywhere else for publication.

Details

Multidiscipline Modeling in Materials and Structures, vol. ahead-of-print no. ahead-of-print
Type: Research Article
ISSN: 1573-6105

Keywords

Click here to view access options
Article
Publication date: 7 August 2017

Ram Jiwari and Ali Saleh Alshomrani

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…

Abstract

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 Runge-Kutta 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) and Kadalbajoo et al. (2005).

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

Click here to view access options
Article
Publication date: 4 April 2020

Neeraj Dhiman, Amit Chauhan, Mohammad Tamsir and Anand Chauhan

A collocation technique based on re-defined quintic B-splines over Crank-Nicolson is presented to solve the Fisher's type equation. We take three cases of aforesaid…

Abstract

Purpose

A collocation technique based on re-defined quintic B-splines over Crank-Nicolson is presented to solve the Fisher's type equation. We take three cases of aforesaid equation. The stability analysis and rate of convergence are also done.

Design/methodology/approach

The quintic B-splines are re-defined which are used for space integration. Taylor series expansion is applied for linearization of the nonlinear terms. The discretization of the problem gives up linear system of equations. A Gaussian elimination method is used to solve these systems.

Findings

Three examples are taken for analysis. The analysis gives guarantee that the present method provides much better results than previously presented methods in literature. The stability analysis and rate of convergence show that the method is unconditionally stable and quadratic convergent for Fisher's type equation. Moreover, the present method is simple and easy to implement, so it may be considered as an alternative method to solve PDEs.

Originality/value

This work is the original work of authors which is neither published nor submitted anywhere else for publication.

Details

Multidiscipline Modeling in Materials and Structures, vol. 16 no. 5
Type: Research Article
ISSN: 1573-6105

Keywords

Click here to view access options
Article
Publication date: 16 October 2018

Rajni Rohila and R.C. Mittal

This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection…

Abstract

Purpose

This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method.

Design/methodology/approach

A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids.

Findings

Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also.

Originality/value

ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 28 no. 11
Type: Research Article
ISSN: 0961-5539

Keywords

Click here to view access options
Article
Publication date: 5 June 2019

Ram Jiwari, Sanjay Kumar and R.C. Mittal

The purpose of this paper is to develop two meshfree algorithms based on multiquadric radial basis functions (RBFs) and differential quadrature (DQ) technique for…

Abstract

Purpose

The purpose of this paper is to develop two meshfree algorithms based on multiquadric radial basis functions (RBFs) and differential quadrature (DQ) technique for numerical simulation and to capture the shocks behavior of Burgers’ type problems.

Design/methodology/approach

The algorithms convert the problems into a system of ordinary differential equations which are solved by the Runge–Kutta method.

Findings

Two meshfree algorithms are developed and their stability is discussed. Numerical experiment is done to check the efficiency of the algorithms, and some shock behaviors of the problems are presented. The proposed algorithms are found to be accurate, simple and fast.

Originality/value

The present algorithms LRBF-DQM and GRBF-DQM are based on radial basis functions, which are new for Burgers’ type problems. It is concluded from the numerical experiments that LRBF-DQM is better than GRBF-DQM. The algorithms give better results than available literature.

Click here to view access options
Article
Publication date: 12 October 2018

Umer Saeed, Mujeeb ur Rehman and Qamar Din

The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more…

Abstract

Purpose

The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more accurate results.

Design/methodology/approach

The authors proposed a method by using the Chebyshev wavelets in conjunction with differential quadrature technique. The operational matrices for the method are derived, constructed and used for the solution of nonlinear fractional partial differential equations.

Findings

The operational matrices contain many zero entries, which lead to the high efficiency of the method and reasonable accuracy is achieved even with less number of grid points. The results are in good agreement with exact solutions and more accurate as compared to Haar wavelet method.

Originality/value

Many engineers can use the presented method for solving their nonlinear fractional models.

Click here to view access options
Article
Publication date: 20 October 2020

Sapna Pandit and R.C. Mittal

This paper aims to propose a novel approach based on uniform scale-3 Haar wavelets for unsteady state space fractional advection-dispersion partial differential equation…

Abstract

Purpose

This paper aims to propose a novel approach based on uniform scale-3 Haar wavelets for unsteady state space fractional advection-dispersion partial differential equation which arises in complex network, fluid dynamics in porous media, biology, chemistry and biochemistry, electrode – electrolyte polarization, finance, system control, etc.

Design/methodology/approach

Scale-3 Haar wavelets are used to approximate the space and time variables. Scale-3 Haar wavelets converts the problems into linear system. After that Gauss elimination is used to find the wavelet coefficients.

Findings

A novel algorithm based on Haar wavelet for two-dimensional fractional partial differential equations is established. Error estimation has been derived by use of property of compactly supported orthonormality. The correctness and effectiveness of the theoretical arguments by numerical tests are confirmed.

Originality/value

Scale-3 Haar wavelets are used first time for these types of problems. Second, error analysis in new work in this direction.

Details

Engineering Computations, vol. 38 no. 4
Type: Research Article
ISSN: 0264-4401

Keywords

Click here to view access options
Article
Publication date: 6 July 2015

R C Mittal and Amit Tripathi

The purpose of this paper is to develop an efficient numerical scheme for non-linear two-dimensional (2D) parabolic partial differential equations using modified bi-cubic…

Abstract

Purpose

The purpose of this paper is to develop an efficient numerical scheme for non-linear two-dimensional (2D) parabolic partial differential equations using modified bi-cubic B-spline functions. As a test case, method has been applied successfully to 2D Burgers equations.

Design/methodology/approach

The scheme is based on collocation of modified bi-cubic B-Spline functions. The authors used these functions for space variable and for its derivatives. Collocation form of the partial differential equation results into system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by strong stability preserving Runge-Kutta method. The computational complexity of the method is O(p log(p)), where p denotes total number of mesh points.

Findings

Obtained numerical solutions are better than those available in literature. Ease of implementation and very small size of computational work are two major advantages of the present method. Moreover, this method provides approximate solutions not only at the grid points but also at any point in the solution domain.

Originality/value

First time, modified bi-cubic B-spline functions have been applied to non-linear 2D parabolic partial differential equations. Efficiency of the proposed method has been confirmed with numerical experiments. The authors conclude that the method provides convergent approximations and handles the equations very well in different cases.

Click here to view access options
Article
Publication date: 26 October 2020

Özlem Ersoy Hepson

The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation.

Abstract

Purpose

The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation.

Design/methodology/approach

The finite elements method (FEM) is a numerical method for obtaining an approximate solution of partial differential equations (PDEs). The development of high-speed computers enables to development FEM to solve PDEs on both complex domain and complicated boundary conditions. It also provides higher-order approximation which consists of a vector of coefficients multiplied by a set of basis functions. FEM with the B-splines is efficient due both to giving a smaller system of algebraic equations that has lower computational complexity and providing higher-order continuous approximation depending on using the B-splines of high degree.

Findings

The result of the test problems indicates the reliability of the method to get solutions to the CBE. QTT B-spline collocation approach has convergence order 3 in space and order 1 in time. So that nonpolynomial splines provide smooth solutions during the run of the program.

Originality/value

There are few numerical methods build-up using the trigonometric tension spline for solving differential equations. The tension B-spline collocation method is used for finding the solution of Coupled Burgers’ equation.

Details

Engineering Computations, vol. 38 no. 5
Type: Research Article
ISSN: 0264-4401

Keywords

Click here to view access options
Article
Publication date: 7 August 2017

Sapna Pandit, Manoj Kumar, R.N. Mohapatra and Ali Saleh Alshomrani

This paper aims to find the numerical solution of planar and non-planar Burgers’ equation and analysis of the shock behave.

Abstract

Purpose

This paper aims to find the numerical solution of planar and non-planar Burgers’ equation and analysis of the shock behave.

Design/methodology/approach

First, the authors discritize the time-dependent term using Crank–Nicholson finite difference approximation and use quasilinearization to linearize the nonlinear term then apply Scale-2 Haar wavelets for space integration. After applying this scheme on partial differential, the equation transforms into a system of algebraic equation. Then, the system of equation is solved using Gauss elimination method.

Findings

Present method is the extension of the method (Jiwari, 2012). The numerical solutions using Scale-2 Haar wavelets prove that the proposed method is reliable for planar and non-planar nonlinear Burgers’ equation and yields results better than other methods and compatible with the exact solutions.

Originality/value

The numerical results for non-planar Burgers’ equation are very sparse. In the present paper, the authors identify where the shock wave and discontinuity occur in planar and non-planar Burgers’' equation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

1 – 10 of over 3000