Legendre Wavelet Method Research Articles

Thermal and velocity slip effects on MHD mixed convection flow of Williamson nanofluid along a vertical surface: Modified Legendre wavelets approach

The purpose of the present work is to carry out the numerical simulation of magnetohydrodynamic (MHD) stagnation-point flow of non-Newtonian Williamson nanofluid towards a vertical moving surface. The focus of present work is to analyze the nanofluid characteristics for both buoyancy assisting and opposing flow regions. The stretching sheet is subject to uniform normal magnetic field and three types of slip conditions: velocity, thermal and solutal slip are applied. The governing equations are derived in the form of nonlinear partial differential equations. Using suitable transformation variables such equations are converted into favorable nonlinear ordinary differential equations. New modification in Legendre wavelets method is proposed and applied to investigate the numerical solution of the reduced set of nonlinear ordinary differential equations. The simulation is performed for the emerging physical parameter on the nanofluid velocity, temperature, concentration, coefficient of skin friction, heat and mass transfer rate. The study concludes that the Modified Legendre Wavelets Method (MLWM) reduces the computational cost and very accurate to find the solution and can be extended for nonlinear diversify problems of physical nature. Moreover, skin friction at the stretching surface is increased due to increase in non-Newtonian Williamson fluid parameter which subsequently decreases the nanofluid velocity in the boundary layer.

Numerical solution of fuzzy fractional integro-differential equation via two-dimensional Legendre Wavelet method

Numerical solution of fuzzy fractional integro-differential equation via two-dimensional Legendre Wavelet method

Applying Legendre wavelet method with Tikhonov regularization for one-dimensional time-fractional diffusion equations

In this paper, a new method based on two-dimensional Legendre wavelet basis is proposed to solve time-fractional diffusion equations. This technique is used to convert the problem into a system of linear algebraic equations via expanding the required approximation based on the Legendre wavelet basis. The effectiveness of the proposed method is examined by comparing the numerical results with other methods. When the desired system is large, it is ill-conditioned; in this case, Tikhonov regularization method, with discrepancy principle for finding the regularization parameter, is applied to stabilize the solution.

Wavelets optimization method for evaluation of fractional partial differential equations: an application to financial modelling

In the present paper, we employ a wavelets optimization method is employed for the elucidations of fractional partial differential equations of pricing European option accompanied by a Lévy model. We apply the Legendre wavelets optimization method (LWOM) to optimize the governing problem. The novelty of the proposed method is the inclusion of differential evolution algorithm (DE) in the Legendre wavelets method for the optimized approximations of the unknown terms of the Legendre wavelets. Sequentially, the functions and components of the pricing models are discretized by utilizing the operational matrix of fractional integration of Legendre wavelets. Illustratively, the implementation of the LWOM is exemplified on a pricing European option Lévy model and successfully depicted the stock paths. Moreover, comparison analysis of the Black-Scholes model with a class of Lévy model and LWOM with q-homotopy analysis transform method (q-HATM) is also deliberated out. Accordingly, the technique is found to be appropriate for financial models that can be expressed as partial differential equations of integer and fractional orders, subjected to initial or boundary conditions.

A semi-discrete scheme for solving fourth-order partial integro-differential equation with a weakly singular kernel using Legendre wavelets method

In this paper, a semi-discrete scheme is presented for solving fourth-order partial integro-differential equations with a weakly singular kernel. The second-order backward difference formula is used to discretize the temporal derivatives. After discretizing the temporal derivatives, the considered problems are converted into a set of ordinary differential equations which are solved by using Legendre wavelets collocation method. The stability and convergence properties related to the time discretization are discussed and theoretically proven. Several numerical examples are included to demonstrate the accuracy and efficiency of the proposed method.

A Numerical Study of Fractional Order Reverse Osmosis Desalination Model using Legendre Wavelet Approximation

The purpose of this study is to develop a new approach in modeling and simulation of a reverse osmosis desalination system by using fractional differential equations. Using the Legendre wavelet method combined with the decoupling and quasi-linearization technique, we demonstrate the validity and applicability of our model. Examples are developed to illustrate the fractional differential technique and to highlight the broad applicability and the efficiency of this method. The fractional derivative is described in the Caputo sense.

A generalized fractional-order Legendre wavelet Tau method for solving fractional differential equations

A new fractional-order wavelet basis as generalization of the classical Legendre wavelet is defined. The operational matrices both for derivative and fractional derivative in the sense of Caputo for this fractional-order wavelet are derived. Then, a numerical scheme based on these operational matrices and the typical Tau method is proposed for solving some nonlinear fractional differential equations. Illustrative examples show that the present wavelet Tau method is numerically effective and convenient for solving fractional differential equations. Moreover, the obtained results confirm that, in comparison with the classical Legendre wavelet method, the fractional-order wavelet basis is more efficient and accurate for solving fractional differential equations. Error analysis and convergence of the proposed wavelet method are also provided.

MHD squeezing flow between two parallel disks with suction or injection via Legendre wavelet-quasilinearization technique

PurposeThis paper aims to propose a method by merging Legendre wavelets method and quasilinearization technique to tackle with the nonlinearity and to get better and more accurate results.Design/methodology/approachTo test the significance of the proposed scheme, the authors applied the method on the model representing magneto-hydrodynamic squeezing flow of a viscous fluid between two parallel infinite disks, where one disk is impermeable and the other is porous with either suction or injection of the fluid. For the sake of comparison, numerical solution by using RK-4 is also computed. From the graphs and tables, it is evident that the proposed method shows an excellent accordance with the numerical solution.FindingsThe solution converges to the numerical solution when the degree of Legendre polynomials m is increased. For m = 20 in all the three cases, for different values of S, M and A, the graphs of solutions obtained by Legendre wavelet quasilinearization technique show an excellent agreement with numerical solution. Also, it is evident from figures that suction and injection affects the velocity profile in opposite way. For suction, maximum velocity is seen to be at the center of the channel. Magnetic field can be used to regularize the flow and it stabilizes the flow behavior.Originality/valueMagnetic field can be used to regularize the flow and it stabilizes the flow behavior.

Application of the Legendre wavelets method to the parallel plate flow of a third grade fluid and forced convection in a porous duct

A method based on Legendre wavelets is presented in this paper to discuss the flow of a third grade fluid between parallel plates and the forced convection in a porous duct. The flow problems are modeled in terms of integral equations which are then solved by the Legendre wavelets method. The comparison between present results and the existing solutions shows that the Legendre wavelets method is a powerful tool for solving nonlinear boundary value problems. We hope this method can be used for solving many interesting problems arising in non-Newtonian fluids.

Two-dimensional Legendre wavelet method for travelling wave solutions of time-fractional generalized seventh order KdV equation

In this paper, a new method based on the Legendre wavelets expansion together with operational matrices of fractional integration and derivative of these basis functions is proposed to solve time-fractional seventh order KdV (sKdV) equation. Two-dimensional Legendre wavelet method is applied to compute the numerical solution of nonlinear time-fractional sKdV equation. The approximate solutions of nonlinear time fractional sKdV equation thus obtained by Legendre wavelet method are compared with the exact solutions as well as homotopy analysis method (HAM). The present scheme is very simple, effective and convenient for obtaining numerical solution of the sKdV equation.

A study of nonlinear biochemical reaction model

The present study deals with the introduction of an alteration in Legendre wavelets method by availing of the Picard iteration method for system of differential equations and named it Legendre wavelet-Picard method (LWPM). Convergence of the proposed method is also discussed. In order to check the competence of the proposed method, basic enzyme kinetics is considered. Systems of nonlinear ordinary differential equations are formed from the considered enzyme-substrate reaction. The results obtained by the proposed LWPM are compared with the numerical results obtained from Runge–Kutta method of order four (RK-4). Numerical results and those obtained by LWPM are in excellent conformance, which would be explained by the help of table and figures. The proposed method is easy and simple to implement as compared to the other existing analytical methods used for solving systems of differential equations arising in biology, physics and engineering.

An efficient approach for solving Klein–Gordon equation arising in quantum field theory using wavelets

A new approach for the solution of Klein–Gordon equation using Legendre wavelet-based approximation method is presented. The properties of Legendre wavelets are used to reduce the problem to the solution of system of algebraic equations. Usually $$2^{(k-1)}M^2$$ connection coefficients are required to solve the Klein–Gordon equation by Legendre wavelets method as mentioned in the literature. But, our proposed method based on Legendre wavelets and algebraic polynomials require only $$2^{(k-1)} M$$ connection coefficients instead of $$2^{(k-1)} M^2$$ . Also the convergence analysis and error estimation for the proposed function approximation through the truncated series have been discussed and approved with the exact solution. Illustrative examples are discussed to demonstrate the validity and applicability of the technique with lesser computational effort.

A new approach for solving a model for HIV infection of $$\mathbf{CD4}^{+}\,\mathbf{T}$$ CD 4 + T -cells arising in mathematical chemistry using wavelets

In this paper, the Legendre wavelet method for solving a model for HIV infection of \(\hbox {CD}4^{+}\,\hbox {T}\)-cells is studied. The properties of Legendre wavelets and its operational matrices are first presented and then are used to convert into algebraic equations. Also the convergence and error analysis for the proposed technique have been discussed. Illustrative examples have been given to demonstrate the validity and applicability of the technique. The efficiency of the proposed method has been compared with other traditional methods and it is observed that the Legendre wavelet method is more convenient than the other methods in terms of applicability, efficiency, accuracy, error, and computational effort.

Legendre wavelets method for the numerical solution of fractional integro-differential equations with weakly singular kernel

In this paper, numerical solutions of the linear and nonlinear fractional integro- differential equations with weakly singular kernel where fractional derivatives are considered in the Caputo sense, have been obtained by Legendre wavelets method. The block pulse functions and their properties are employed to derive a general procedure for forming the operational matrix of fractional integration for Legendre wavelets. The application of this matrix for solving initial problem is explained. The mentioned equations are transformed into a system of algebraic equations. The error analysis of the proposed method is investigated. Finally, some numerical examples are shown to illustrate the efficiency of the approach.

Modified Legendre Wavelets Technique for Fractional Oscillation Equations

Physical Phenomena’s located around us are primarily nonlinear in nature and their solutions are of highest significance for scientists and engineers. In order to have a better representation of these physical models, fractional calculus is used. Fractional order oscillation equations are included among these nonlinear phenomena’s. To tackle with the nonlinearity arising, in these phenomena’s we recommend a new method. In the proposed method, Picard’s iteration is used to convert the nonlinear fractional order oscillation equation into a fractional order recurrence relation and then Legendre wavelets method is applied on the converted problem. In order to check the efficiency and accuracy of the suggested modification, we have considered three problems namely: fractional order force-free Duffing–van der Pol oscillator, forced Duffing–van der Pol oscillator and higher order fractional Duffing equations. The obtained results are compared with the results obtained via other techniques.

Hybrid Legendre Block-Pulse functions for the numerical solutions of system of nonlinear Fredholm–Hammerstein integral equations

In this paper, the numerical technique based on hybrid Legendre-Block-Pulse function has been developed to approximate the solution of system of nonlinear Fredholm–Hammerstein integral equations. These functions are formed by the hybridization of Legendre polynomials and Block-Pulse functions. These functions are orthonormal and have compact support on [0,1]. The proposed method reduces the system of integral equations to a system of nonlinear algebraic equations that can be solved easily by any usual numerical method. The numerical results obtained by the presented method have been compared with those obtained by Legendre wavelet method (LWM). Numerical examples are presented to illustrate the accuracy of the method.

Numerical Solution of Fractional Partial Differential Equation of Parabolic Type With Dirichlet Boundary Conditions Using Two-Dimensional Legendre Wavelets Method

In this paper, the numerical solution for the fractional order partial differential equation (PDE) of parabolic type has been presented using two dimensional (2D) Legendre wavelets method. 2D Haar wavelets method is also applied to compute the numerical solution of nonlinear time-fractional PDE. The approximate solutions of nonlinear fractional PDE thus obtained by Haar wavelet method and Legendre wavelet method are compared with the exact solution obtained by using homotopy perturbation method (HPM). The present scheme is simple, effective, and expedient for obtaining numerical solution of the fractional PDE.

SOLUTION OF HIGHER ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS BY LEGENDRE WAVELETS

In this paper the solution of higher order Volterra integro-differential equations using Legendre wavelets is presented. The problem is reduced into the solution of non-linear algebraic equations using Legendre wavelets. A reliable approach is also discussed for the convergence of the Legendre wavelets. To demonstrate the validity and applicability of the proposed method, some examples have been discussed. The results in these examples demonstrate the reliability and efficiency of the proposed method. The results obtained by the Legendre wavelet method are very near to exact solution. AMS Subject Classification: 42C40, 42C10, 45J05

State analysis of time-varying singular nonlinear systems using Legendre wavelets

In this paper, the Legendre wavelet method for State analysis of time-varying singular nonlinear systems is studied. The properties of Legendre wavelets and its operational matrices are first presented and then are used to convert into algebraic equations. Also the convergence and error analysis for the proposed technique have been discussed. Illustrative examples have been given to demonstrate the validity and applicability of the technique. The efficiency of the proposed method has been compared with Haar wavelet method and it is observed that the Legendre wavelet method is more convenient than the Haar wavelet method in terms of applicability, efficiency, accuracy, error, and computational effort.

Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system

In this paper, Legendre wavelet method is developed to approximate the solutions of system of nonlinear Volterra integro-differential equations. The properties of Legendre wavelets are first presented. The properties of Legendre wavelets are used to reduce the system of integral equations to a system of algebraic equations which can be solved numerically by Newton’s method. Also, the results obtained by present method have been compared with that of by B-spline wavelet method. Illustrative examples have been discussed to demonstrate the validity and applicability of the present method.