Search results

1 – 10 of 167
Article
Publication date: 8 August 2020

Amit K. Verma, Narendra Kumar and Diksha Tiwari

The purpose of this paper is to propose an efficient computational technique, which uses Haar wavelets collocation approach coupled with the Newton-Raphson method and solves the…

Abstract

Purpose

The purpose of this paper is to propose an efficient computational technique, which uses Haar wavelets collocation approach coupled with the Newton-Raphson method and solves the following class of system of Lane–Emden equations:

(tk1y(t))=tω1f1(t,y(t),z(t)),
(tk2z(t))=tω2f2(t,y(t),z(t)),
where t > 0, subject to the following initial values, boundary values and four-point boundary values:
y(0)=γ1, y(0)=0, z(0)=γ2, z(0)=0,
y(0)=0, y(1)=δ1, z(0)=0, z(1)=δ2,
y(0)=0, y(1)=n1z(v1), z(0)=0, z(1)=n2y(v2),
where n1,n2,v1,v2(0,1) and k10,k20,ω1<1,ω2<1, γ1, γ2, δ1, δ2 are real constants.

Design/methodology/approach

To deal with singularity, Haar wavelets are used, and to deal with the nonlinear system of equations that arise during computation, the Newton-Raphson method is used. The convergence of these methods is also established and the results are compared with existing techniques.

Findings

The authors propose three methods based on uniform Haar wavelets approximation coupled with the Newton-Raphson method. The authors obtain quadratic convergence for the Haar wavelets collocation method. Test problems are solved to validate various computational aspects of the Haar wavelets approach. The authors observe that with only a few spatial divisions the authors can obtain highly accurate solutions for both initial value problems and boundary value problems.

Originality/value

The results presented in this paper do not exist in the literature. The system of nonlinear singular differential equations is not easy to handle as they are singular, as well as nonlinear. To the best of the knowledge, these are the first results for a system of nonlinear singular differential equations, by using the Haar wavelets collocation approach coupled with the Newton-Raphson method. The results developed in this paper can be used to solve problems arising in different branches of science and engineering.

Details

Engineering Computations, vol. 38 no. 2
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 24 August 2018

R.C. Mittal and Sapna Pandit

The main purpose of this work is to develop a novel algorithm based on Scale-3 Haar wavelets (S-3 HW) and quasilinearization for numerical simulation of dynamical system of…

Abstract

Purpose

The main purpose of this work is to develop a novel algorithm based on Scale-3 Haar wavelets (S-3 HW) and quasilinearization for numerical simulation of dynamical system of ordinary differential equations.

Design/methodology/approach

The first step in the development of the algorithm is quasilinearization process to linearize the problem, and then Scale-3 Haar wavelets are used for space discretization. Finally, the obtained system is solved by Gauss elimination method.

Findings

Some numerical examples of fractional dynamical system are considered to check the accuracy of the algorithm. Numerical results show that quasilinearization with Scale-3 Haar wavelet converges fast even for small number of collocation points as compared of classical Scale-2 Haar wavelet (S-2 HW) method. The convergence analysis of the proposed algorithm has been shown that as we increase the resolution level of Scale-3 Haar wavelet error goes to zero rapidly.

Originality/value

To the best of authors’ knowledge, this is the first time that new Haar wavelets Scale-3 have been used in fractional system. A new scheme is developed for dynamical system based on new Scale-3 Haar wavelets. These wavelets take less time than Scale-2 Haar wavelets. This approach extends the idea of Jiwari (2015, 2012) via translation and dilation of Haar function at Scale-3.

Details

Engineering Computations, vol. 35 no. 5
Type: Research Article
ISSN: 0264-4401

Keywords

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 which…

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

Article
Publication date: 23 November 2020

Gopal Priyadarshi and B.V. Rathish Kumar

In the past few years, Haar wavelet-based numerical methods have been applied successfully to solve linear and nonlinear partial differential equations. This study aims to propose…

Abstract

Purpose

In the past few years, Haar wavelet-based numerical methods have been applied successfully to solve linear and nonlinear partial differential equations. This study aims to propose a wavelet collocation method based on Haar wavelets to identify a parameter in parabolic partial differential equations (PDEs). As Haar wavelet is defined in a very simple way, implementation of the Haar wavelet method becomes easier than the other numerical methods such as finite element method and spectral method. The computational time taken by this method is very less because Haar matrices and Haar integral matrices are stored once and used for each iteration. In the case of Haar wavelet method, Dirichlet boundary conditions are incorporated automatically. Apart from this property, Haar wavelets are compactly supported orthonormal functions. These properties lead to a huge reduction in the computational cost of the method.

Design/methodology/approach

The aim of this paper is to reconstruct the source control parameter arises in quasilinear parabolic partial differential equation using Haar wavelet-based numerical method. Haar wavelets possess various properties, for example, compact support, orthonormality and closed form expression. The main difficulty with the Haar wavelet is its discontinuity. Therefore, this paper cannot directly use the Haar wavelet to solve partial differential equations. To handle this difficulty, this paper represents the highest-order derivative in terms of Haar wavelet series and using successive integration this study obtains the required term appearing in the problem. Taylor series expansion is used to obtain the second-order partial derivatives at collocation points.

Findings

An efficient and accurate numerical method based on Haar wavelet has been proposed for parameter identification in quasilinear parabolic partial differential equations. Numerical results are obtained from the proposed method and compared with the existing results obtained from various finite difference methods including Saulyev method. It is shown that the proposed method is superior than the conventional finite difference methods including Saulyev method in terms of accuracy and CPU time. Convergence analysis is presented to show the accuracy of the proposed method. An efficient algorithm is proposed to find the wavelet coefficients at target time.

Originality/value

The outcome of the paper would have a valuable role in the scientific community for several reasons. In the current scenario, the parabolic inverse problem has emerged as very important problem because of its application in many diverse fields such as tomography, chemical diffusion, thermoelectricity and control theory. In this paper, higher-order derivative is represented in terms of Haar wavelet series. In other words, we represent the solution in multiscale framework. This would enable us to understand the solution at various resolution levels. In the case of Haar wavelet, this paper can achieve a very good accuracy at very less resolution levels, which ultimately leads to huge reduction in the computational cost.

Details

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

Keywords

Article
Publication date: 5 January 2015

Manoj Kumar and Sapna Pandit

The purpose of this paper is to discuss the application of the Haar wavelets for solving linear and nonlinear Fokker-Planck equations with appropriate initial and boundary…

Abstract

Purpose

The purpose of this paper is to discuss the application of the Haar wavelets for solving linear and nonlinear Fokker-Planck equations with appropriate initial and boundary conditions.

Design/methodology/approach

Haar wavelet approach converts the problems into a system of linear algebraic equations and the obtained system is solved by Gauss-elimination method.

Findings

The accuracy of the proposed scheme is demonstrated on three test examples. The numerical solutions prove that the proposed method is reliable and yields compatible results with the exact solutions. The scheme provides better results than the schemes [9, 14].

Originality/value

The developed scheme is a new scheme for Fokker-Planck equations. The scheme based on Haar wavelets is expended for nonlinear partial differential equations with variable coefficients.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 5 June 2017

Ram Jiwari, Vikas Kumar, Ram Karan and Ali Saleh Alshomrani

This paper aims to deal with two-dimensional magneto-hydrodynamic (MHD) Falkner–Skan boundary layer flow of an incompressible viscous electrically conducting fluid over a…

Abstract

Purpose

This paper aims to deal with two-dimensional magneto-hydrodynamic (MHD) Falkner–Skan boundary layer flow of an incompressible viscous electrically conducting fluid over a permeable wall in the presence of a magnetic field.

Design/methodology/approach

Using the Lie group approach, the Lie algebra of infinitesimal generators of equivalence transformations is constructed for the equation under consideration. Using these suitable similarity transformations, the governing partial differential equations are reduced to linear and nonlinear ordinary differential equations (ODEs). Further, Haar wavelet approach is applied to the reduced ODE under the subalgebra 4.1 for constructing numerical solutions of the flow problem.

Findings

A new type of solutions was obtained of the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis.

Originality/value

To find a solution for the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis is a new approach for fluid problems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 6 November 2017

Sapna Pandit, Ram Jiwari, Karan Bedi and Mehmet Emir Koksal

The purpose of this study is to develop an algorithm for approximate solutions of nonlinear hyperbolic partial differential equations.

Abstract

Purpose

The purpose of this study is to develop an algorithm for approximate solutions of nonlinear hyperbolic partial differential equations.

Design/methodology/approach

In this paper, an algorithm based on the Haar wavelets operational matrix for computational modelling of nonlinear hyperbolic type wave equations has been developed. These types of equations describe a variety of physical models in nonlinear optics, relativistic quantum mechanics, solitons and condensed matter physics, interaction of solitons in collision-less plasma and solid-state physics, etc. The algorithm reduces the equations into a system of algebraic equations and then the system is solved by the Gauss-elimination procedure. Some well-known hyperbolic-type wave problems are considered as numerical problems to check the accuracy and efficiency of the proposed algorithm. The numerical results are shown in figures and Linf, RMS and L2 error forms.

Findings

The developed algorithm is used to find the computational modelling of nonlinear hyperbolic-type wave equations. The algorithm is well suited for some well-known wave equations.

Originality/value

This paper extends the idea of one dimensional Haar wavelets algorithms (Jiwari, 2015, 2012; Pandit et al., 2015; Kumar and Pandit, 2014, 2015) for two-dimensional hyperbolic problems and the idea of this algorithm is quite different from the idea for elliptic problems (Lepik, 2011; Shi et al., 2012). Second, the algorithm and error analysis are new for two-dimensional hyperbolic-type problems.

Details

Engineering Computations, vol. 34 no. 8
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 3 August 2021

Zain ul Abdeen and Mujeeb ur Rehman

The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.

Abstract

Purpose

The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.

Design/methodology/approach

The aim of this paper is to develop a numerical scheme for numerical solutions of Hadamard-type fractional differential equations. The classical Haar wavelets are modified to align them with Hadamard-type operators. Operational matrices are derived and used to convert differential equations to systems of algebraic equations.

Findings

The upper bound for error is estimated. With the help of quasilinearization, nonlinear problems are converted to sequences of linear problems and operational matrices for modified Haar wavelets are used to get their numerical solution. Several numerical examples are presented to demonstrate the applicability and validity of the proposed method.

Originality/value

The numerical method is purposed for solving Hadamard-type fractional differential equations.

Details

Engineering Computations, vol. 39 no. 3
Type: Research Article
ISSN: 0264-4401

Keywords

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

Article
Publication date: 2 June 2023

Dhabaleswar Mohapatra and Snehashish Chakraverty

Investigation of the smoking model is important as it has a direct effect on human health. This paper focuses on the numerical analysis of the fractional order giving up smoking…

Abstract

Purpose

Investigation of the smoking model is important as it has a direct effect on human health. This paper focuses on the numerical analysis of the fractional order giving up smoking model. Nonetheless, due to observational or experimental errors, or any other circumstance, it may contain some incomplete information. Fuzzy sets can be used to deal with uncertainty. Yet, there may be some inconsistency in the membership as well. As a result, the primary goal of this proposed work is to numerically solve the model in a type-2 fuzzy environment.

Design/methodology/approach

Triangular perfect quasi type-2 fuzzy numbers (TPQT2FNs) are used to deal with the uncertainty in the model. In this work, concepts of r2-cut at r1-plane are used to model the problem's uncertain parameter. The Legendre wavelet method (LWM) is then utilised to solve the giving up smoking model in a type-2 fuzzy environment.

Findings

LWM has been effectively employed in conjunction with the r2-cut at r1-plane notion of type-2 fuzzy sets to solve the model. The LWM has the advantage of converting the non-linear fractional order model into a set of non-linear algebraic equations. LWM scheme solutions are found to be well agreed with RK4 scheme solutions. The existence and uniqueness of the model's solution have also been demonstrated.

Originality/value

To deal with the uncertainty, type-2 fuzzy numbers are used. The use of LWM in a type-2 fuzzy uncertain environment to achieve the model's required solutions is quite fascinating, and this is the key focus of this work.

Details

Engineering Computations, vol. 40 no. 4
Type: Research Article
ISSN: 0264-4401

Keywords

1 – 10 of 167