Quasilinearization Process Research Articles

A Second-Order Scheme for the Generalized Time-Fractional Burgers' Equation

Abstract A second-order numerical scheme is proposed to solve the generalized time-fractional Burgers' equation. The time-fractional derivative is considered in the Caputo sense. First, the quasi-linearization process is used to linearize the time-fractional Burgers' equation, which gives a sequence of linear partial differential equations (PDEs). The Crank–Nicolson scheme is used to discretize the sequence of PDEs in the temporal direction, followed by the central difference formulae for both the first and second-order spatial derivatives. The established error bounds (in the L2− norm) obtained through the meticulous theoretical analysis show that the method is second-order convergent in space and time. The technique is also shown to be conditionally stable. Some numerical experiments are presented to confirm the theoretical results.

Meshfree algorithms for analysis and computational modeling of multidimensional hyperbolic wave models

PurposeThis article offered two meshfree algorithms, namely the local radial basis functions-finite difference (LRBF-FD) approximation and local radial basis functions-differential quadrature method (LRBF-DQM) to simulate the multidimensional hyperbolic wave models and work is an extension of Jiwari (2015).Design/methodology/approachIn the evolvement of the first algorithm, the time derivative is discretized by the forward FD scheme and the Crank-Nicolson scheme is used for the rest of the terms. After that, the LRBF-FD approximation is used for spatial discretization and quasi-linearization process for linearization of the problem. Finally, the obtained linear system is solved by the LU decomposition method. In the development of the second algorithm, semi-discretization in space is done via LRBF-DQM and then an explicit RK4 is used for fully discretization in time.FindingsFor simulation purposes, some 1D and 2D wave models are pondered to instigate the chastity and competence of the developed algorithms.Originality/valueThe developed algorithms are novel for the multidimensional hyperbolic wave models. Also, the stability analysis of the second algorithm is a new work for these types of model.

A high-order reliable and efficient Haar wavelet collocation method for nonlinear problems with two point-integral boundary conditions

The primary goal of this study is to increase and improve the precision and order of convergence of the well-known Haar wavelet collocation method (HWCM) that is named as Higher order Haar wavelet collocation method (HHWCM). The HHWCM will then be applied to nonlinear ordinary differential equations with a variety of initial conditions, boundary conditions, periodic conditions, two-point conditions, integral conditions, and multi-point integral boundary conditions. The paper also contains important theorem about the convergence of the HHWCM with computational stability. The convergence of HHWCM is then compared to recently published works including the famous HWCM. In nonlinear case the quasilinearization process has been introduced to linearize the differential equation. Different orders of differential equations including homogeneous and non–homogeneous equations with constant and variable coefficients are also tested by HHWCM. The key advantages of the HHWCM include its easy to use, stability, convergence, high order accuracy, and efficiency under a range of boundary conditions. We have also implemented the HHWCM on nonlinear differential equation having no exact solution.

Study of 4th order Kuramoto-Sivashinsky equation by septic Hermite collocation method

In the current study, a robust septic Hermite collocation method (SHCM) is proposed to simulate the Kuramoto-Sivashinsky (KS) equation. To approximate the spatial derivatives, the collocation method with the 7th-degree Hermite interpolation polynomial is utilized and for the temporal derivative, the Crank-Nicolson (CN) technique is implemented. The nonlinear terms of the KS equation are linearized using the quasi-linearization process. Using the von-Neumann approach, it is demonstrated that the algorithm is unconditionally stable. The convergence analysis of the proposed technique is also given. In the temporal direction, the scheme is observed to be second-order convergent and in space direction, it is found to be sixth-order convergent. The proposed technique's robustness is demonstrated by solving six test problems. The L2, L∞, and global relative errors are determined and the findings are compared with other methods available in the literature. The behavior of few KS equations, for which their exact solution is not available, is also analyzed. The current findings are better than the results obtained from the other methods and are also matched well with the analytical solution.

Analyze the Flow Behavior for MHD Power-Law Fluids

Objective: This study investigates the Numerical solution of laminar boundary layer flow of Magnetohydrodynamics (MHD) model for power-law fluid over a continuous moving surface in the presence of a transverse magnetic. Methods: The governing partial differential equation for the flow was transformed into non-linear ordinary differential equation using the Group theoretic method. Firstly, we convert this non-linear ordinary differential equation (ODE) into linear by using quasilinearization process. This linear ODE was solved numerically by applying the Spline collocation method suggested by Bickley. Findings: The solution for displacement profile and velocity profile were obtained as functions of the magnetic parameters. The effect of the magnetic parameters was discussed graphically. We used MATLAB software for finding the outcomes. Novelty: The main goal of this article is to analyze boundary layer flow of Magneto hydrodynamics (MHD) model for power-law fluid over a continuous moving surface in the presence of a transverse magnetic. The conservation equations of mass, momentum and energy are converted into ordinary differential equations along with boundary conditions by appropriate similarity transformations and solved by applying Spline Collocation Method. The convergence of solutions is important for providing the developing linear functions of solutions, which is a benefit of the Spline Collocation Method. These research findings are applicable, for example, in predicting skin friction and heat transfer rate over a stretching sheet, which has implications in technological and manufacturing industries such as polymer extrusion. Comparisons with previously published works are made, and the results show a high level of agreement. This type of research is applicable to work in fire dynamics in insulation, solar collection systems, recovery of petroleum products, etc. Keywords: Power-Law Fluids; Magnetic Field; Nonlinear Differential Equation; Quasilinearization; Bickley’s Method; Linear Equations

A New Parameter-Uniform Discretization of Semilinear Singularly Perturbed Problems

In this paper, we present a numerical approach to solving singularly perturbed semilinear convection-diffusion problems. The nonlinear part of the problem is linearized via the quasilinearization technique. We then design and implement a fitted operator finite difference method to solve the sequence of linear singularly perturbed problems that emerges from the quasilinearization process. We carry out a rigorous analysis to attest to the convergence of the proposed procedure and notice that the method is first-order uniformly convergent. Some numerical evaluations are implemented on model examples to confirm the proposed theoretical results and to show the efficiency of the method.

Investigation for the numerical solution of Klein-Gordon equations using scale 3 Haar wavelets

In this article, a new numerical technique based upon Haar scale 3 wavelets is proposed for the solution of Klein-Gordon equations. Klein-Gordon equation occurs in many phenomena of applied physics such as quantum field theory, fluid dynamics, etc. Wavelets have become a powerful tool in almost all areas of engineering and science. Therefore, in this technique, spatial discretization is done by Haar scale 3 Wavelets. The collocation method is used with Haar scale 3 wavelets to convert Klein-Gordon equation into the system of algebraic equations. To deal with the non-linearities in equations, the quasilinearization process is used. The proposed scheme is tested on three test problems of the above-said types, and the proposed scheme’s competency is established.

Wavelet Operational Matrices and Lagrange Interpolation Differential Quadrature‐Based Numerical Algorithms for Simulation of Nanofluid in Porous Channel

This work analyses the features of nanofluid flow and thermal transmission (NFTT) in a rectangular channel which is asymmetric by developing two numerical algorithms based on scale‐2 Haar wavelets (S2HWs), Lagrange’s interpolation differential quadrature technique (LIDQT), and quasilinearization process (QP). In the simulation procedure, first of all, using similarity transformation (ST), the governing unsteady 2D flow model is changed into two highly non‐linear ODEs. After that, QP is applied to linearize the non‐linear ODEs, and finally S2HWs and LIDQT are used to simulate the non‐linear system of ODEs. In results and discussion section, the parameters Reynolds number (R), expansion ratio and Nusselt (Nu), and nanoparticle volume fraction (φ) are analysed with respect to velocity and temperature profiles. The proposed techniques are easy to implement for fluid and heat transfer (FHT) problems.

Two different temporal domain integration schemes combined with compact finite difference method to solve modified Burgers’ equation

The modified Burgers’ equation has application in various fields such as explosions and sonic boom theory, propagation of torsional waves in thin circular viscoelastic rod, acoustic waves produced by laser radiations, etc. A sixth-order compact finite difference scheme has been used for the space integration. Two different approaches have been followed for time integration. One is the Crank-Nicolson scheme in which the nonlinear term is handled with a quasilinearization process. Another is the strong stability preserving Runge-Kutta method of four stage, third-order convergence (SSP-RK43), in which no linearization is required. The stability analysis for both the schemes has been presented and is shown to be stable. Few test problems have been solved, demonstrating the performance of the proposed techniques. The L2 and L∞ error norms are calculated and are compared with the previous work. The aim of the paper is to solve such an important equation with better accuracy and less computational efforts.

Stability estimates for singularly perturbed Fisher's equation using element-free Galerkin algorithm

<abstract><p>In the present article, a mesh-free technique has been presented to study the behavior of nonlinear singularly perturbed Fisher's problem, which exhibits the traveling wave propagation phenomenon. Some narrow regions adjacent to the left and right lateral boundary may possess rapid variations when the singular perturbation parameter $ \epsilon\rightarrow 0 $, which are not captured nicely by the traditional numerical schemes. In the current work, a robust numerical strategy is proposed, which comprises the implicit Crank-Nicolson scheme to discretize the time derivative term and the element-free Galerkin (EFG) scheme to discretize the spatial derivative terms with nodes densely distributed in the boundary layer regions. The stability of the semi-discrete scheme has been analyzed, and the rate of convergence is shown to be $ \mathcal{O}(\tau^{2}) $. The nonlinear nature of the considered problem has been tackled by employing the quasilinearization process, and its convergence rate has been discussed. Some numerical experiments have been performed to verify the computational uniformity and robustness of the suggested method, rate of convergence as well $ L_{\infty} $ errors have been presented, which depicts the effectiveness of the proposed method.</p></abstract>

Wavelet-Based Numerical Solution for MHD Boundary-Layer Flow Due to Stretching Sheet

In this paper, a two-dimensional steady flow of a viscous fluid due to a stretching sheet in the presence of a magnetic field is considered. We proposed two new numerical schemes based on the Haar wavelet coupled with a collocation approach and quasi-linearization process for solving the Falkner-Skan equation representing the governing problem. The important derived quantities representing the fluid velocity and wall shear stress for various values of flow parameters <i>M</i> and β are calculated. The proposed methods enable us to obtain the solutions even for negative β, nonlinear stretching parameter, and smaller values of the magnetic parameter (<i>M</i> < <i>1</i>) which was missing in the earlier findings. Numerical and graphical results obtained show an excellent agreement with the available findings and demonstrate the efficiency and accuracy of the developed schemes. Another significant advantage of the present method is that it does not depends on small parameters and initial presumptions unlike in traditional semi-analytical and numerical methods.

Numerical treatment of Benjamin-Bona-Mahony-Burgers equation with fourth-order improvised B-spline collocation method

In this work, the Benjamin-Bona-Mahony-Burgers (BBMB) equation is solved using an improvised cubic B-spline collocation technique. This equation describes the propagation of small amplitude waves in a nonlinear dispersive medium, in the modeling of unidirectional planar waves. Due to the higher smoothness and sparse nature of matrices corresponding to splines, cubic B-splines are chosen as the basis function in the collocation method. But, the optimal accuracy and order of convergence cannot be achieved using the standard B-spline collocation method. So to overcome this, improvised cubic B-splines are formed by making posteriori corrections to cubic B-spline interpolant and its higher-order derivatives. The Crank-Nicolson scheme is used to discretize the temporal domain along with the quasilinearization process to deal with the nonlinear terms. The spatial domain discretization is carried out using the improvised cubic B-spline collocation method (ICSCM). The stability analysis of the technique is performed using the von-Neumann scheme. Several test problems are solved numerically and obtained results are compared with the results available in the literature. The aim of the paper is to show that such improvised techniques which were earlier used to solve ODEs, can be applied to solve the BBMB equation also, with excellent accuracy in results.

An improvised collocation algorithm with specific end conditions for solving modified Burgers equation

AbstractIn this work, numerical solution of nonlinear modified Burgers equation is obtained using an improvised collocation technique with cubic B‐spline as basis functions. In this technique, cubic B‐splines are forced to satisfy the interpolatory condition along with some specific end conditions. Crank–Nicolson scheme is used for temporal domain and improvised cubic B‐spline collocation method is used for spatial domain discretization. Quasilinearization process is followed to tackle the nonlinear term in the equation. Convergence of the technique is established to be of order O(h4 + Δt2). Stability of the technique is examined using von‐Neumann analysis. L2 and L∞ error norms are calculated and are compared with those available in existing works. Results are found to be better and the technique is computationally efficient, which is shown by calculating CPU time.

An exponentially fitted numerical technique for singularly perturbed Burgers-Fisher equation on a layer adapted mesh

ABSTRACTThe main aim of this paper is to propose a robust and efficient numerical scheme for solving time dependent singularly perturbed Burgers-Fisher problem based on mesh adaptive strategy in the finite element framework. The Burgers-Fisher equation in one space dimension is a partial differential equation which exhibits travelling wave phenomenon. Since Burgers-Fisher equation is a non-linear problem, quasilinearization process has been used to deal with nonlinearity occurring in the problem. Time discretization has been performed using implicit Euler method. Then spatial discretization has been carried out using finite element technique based on exponentially fitted splines on piecewise uniform Shishkin mesh. The stability of the proposed numerical scheme has been discussed. At the end, it has been shown numerically that the proposed method is very much effective for capturing sharp boundary layers arising in the solution as singular perturbation parameter.

Quasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynamical systems

PurposeThe 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/approachThe 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.FindingsSome 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/valueTo 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.

Numerical Method for a Class of Nonlinear Singularly Perturbed Delay Differential Equations Using Parametric Cubic Spline

Abstract In this paper, we study the numerical method for a class of nonlinear singularly perturbed delay differential equations using parametric cubic spline. Quasilinearization process is applied to convert the nonlinear singularly perturbed delay differential equations into a sequence of linear singularly perturbed delay differential equations. When the delay is not sufficiently smaller order of the singular perturbation parameter, the approach of expanding the delay term in Taylor’s series may lead to bad approximation. To handle the delay term, we construct a special type of mesh in such a way that the term containing delay lies on nodal points after discretization. The parametric cubic spline is presented for solving sequence of linear singularly perturbed delay differential equations. The error analysis of the method is presented and shows second-order convergence. The effect of delay parameter on the boundary layer behavior of the solution is discussed with two test examples.

A differential quadrature based numerical method for highly accurate solutions of Burgers' equation

In this article, we introduce a new, simple, and accurate computational technique for one‐dimensional Burgers' equation. The idea behind this method is the use of polynomial based differential quadrature (PDQ) for the discretization of both time and space derivatives. The quasilinearization process is used for the elimination of nonlinearity. The resultant scheme has simulated for five classic examples of Burgers' equation. The simulation outcomes are validated through comparison with exact and secondary data in the literature for small and large values of kinematic viscosity. The article has deduced that the proposed scheme gives very accurate results even with less number of grid points. The scheme is found to be very simple to implement. Hence, it applies to any domain requires quick implementation and computation.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2023–2042, 2017

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

PurposeThe 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/approachDirichlet 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.FindingsA 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/valueTo 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).

HAAR WAVELETS BASED TIME DISCRETIZATION TECHNIQUE FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

Abstract: A new numerical technique is developed to find the solutions of general nonlinear partial differential equations. The technique is based on the time discretization of Haar wavelet series approximations with quasilinearization process. In order to test the efficiency of the proposed technique, it is applied on well known nonlinear partial differential equations such as the generalized regularized long wave equation, the Benjamin Bona-Mahony equation and the Fitzhugh-Nagumo equation. Numerical results are obtained by preparing MATLAB codes of proposed techniques. The beautiful concentration profiles of u and v are shown by figures at different time level and error norms L2 and L∞ are calculated.

Haar wavelet solutions of nonlinear oscillator equations

In this paper, we present a numerical scheme using uniform Haar wavelet approximation and quasilinearization process for solving some nonlinear oscillator equations. In our proposed work, quasilinearization technique is first applied through Haar wavelets to convert a nonlinear differential equation into a set of linear algebraic equations. Finally, to demonstrate the validity of the proposed method, it has been applied on three type of nonlinear oscillators namely Duffing, Van der Pol, and Duffing–van der Pol. The obtained responses are presented graphically and compared with available numerical and analytical solutions found in the literature. The main advantage of uniform Haar wavelet series with quasilinearization process is that it captures the behavior of the nonlinear oscillators without any iteration. The numerical problems are considered with force and without force to check the efficiency and simple applicability of method on nonlinear oscillator problems.