Two-dimensional Legendre Wavelet Research Articles

Numerical Solution of Time Fractional Black–Scholes Model Based on Legendre Wavelet Neural Network with Extreme Learning Machine

In this paper, the Legendre wavelet neural network with extreme learning machine is proposed for the numerical solution of the time fractional Black–Scholes model. In this way, the operational matrix of the fractional derivative based on the two-dimensional Legendre wavelet is derived and employed to solve the European options pricing problem. This scheme converts this problem into the calculation of a set of algebraic equations. The Legendre wavelet neural network is constructed; meanwhile, the extreme learning machine algorithm is adopted to speed up the learning rate and avoid the over-fitting problem. In order to evaluate the performance of this scheme, a comparative study with the implicit differential method is constructed to validate its feasibility and effectiveness. Experimental results illustrate that this scheme offers a satisfactory numerical solution compared to the benchmark method.

Numerical analysis of variable fractional viscoelastic column based on two-dimensional Legendre wavelets algorithm

In this paper, two-dimensional Legendre wavelets algorithm is applied for the first time to solve a variable order fractional partial differential governing equation of viscoelastic column in the time domain. Firstly, the governing equation of a viscoelastic column is established according to a variable order fractional constitutive model. Secondly, the unknown function is expanded into the elements of two-dimensional Legendre wavelets. In order to obtain the numerical solutions of this type of equation, the differential operator matrices based on Legendre wavelets of integer order and variable order fractional are derived. The operator matrices are used to convert the initial governing equation into algebraic equations that are easy to solve in the time domain. The efficiency and accuracy of the algorithm are verified through the convergence analysis of Legendre wavelets and the error estimations of numerical example. Finally, the displacement solutions of the viscoelastic column under constant load and variable load are considered, and the columns with different cross-section shapes are studied. The results show that the proposed numerical algorithm is efficient in dynamic analysis of viscoelastic columns.

Approximation of functions of Lipschitz class and solution of Fokker-Planck equation by two-dimensional Legendre wavelet operational matrix

In this paper, Lipschitz class of two-variables is considered. This is the genralization of well-known Lipschitz class of functions. A new estimator of functions belonging to generalized Lipschitz class has been obtained. Also, the solutions for the Fokker-Planck equations have been obtained for two different cases by two-dimensional Legendre wavelet operational matrix method. The approximated solutions of the time-and space-Fokker Planck equation have been compared with the exact solutions and the solutions obtained by homotopy perturbation method. The proposed scheme is simple, effective and suitable for the solution of Fokker-Planck equation.

Numerical wavelets scheme to complex partial differential equation arising from Morlet continuous wavelet transform

AbstractIn this article, a new wavelets approximation scheme based on operational matrices has been discussed for the solutions of proposed initial value complex partial differential equation (CPDE) which is arising from continuous wavelet transform. We developed an algorithm using transformation to obtain the numerical solution of CPDE with a numerical wavelet collocation method based on two‐dimensional Legendre wavelet (TDLW) and two‐dimensional Bernoulli wavelet (TDBW) operational matrices. We converted CPDE into a system of partial differential equations (PDEs) and this system of PDEs converted into couple PDEs in terms of the real and imaginary component. Finally, we obtained a system of partial integro‐differential equations (PIDEs) with nonlocal boundary conditions. After implementation of wavelets scheme on PIDEs, PIDEs reduced into a system of algebraic equations. In the sequence of this, we also investigated the convergence analysis of TDLW, TDBW, and error analysis, respectively. Finally, illustrative examples are included to demonstrate the validity and applicability of the presented wavelet scheme for the proposed problem.

A wavelet method for nonlinear variable-order time fractional 2D Schrödinger equation

<p style='text-indent:20px;'>In this study, an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions for nonlinear variable-order time fractional two-dimensional (2D) Schrödinger equation. First, the variable-order time fractional derivative involved in the considered problem is approximated via the finite difference technique. Then, by help of the finite difference scheme and the theta-weighted method, a recursive algorithm is derived for the problem under examination. After that, the real functions available in the real and imaginary parts of the unknown solution of the problem are expanded via the 2D LWs. Finally, by applying the operational matrices of derivative, the solution of the problem is transformed to the solution of a linear system of algebraic equations in each time step which can simply be solved. In the proposed method, acceptable approximate solutions are achieved by employing only a small number of the basis functions. To illustrate the applicability, validity and accuracy of the wavelet method, some numerical test examples are solved using the suggested method. The achieved numerical results reveal that the method established based on the 2D LWs is very easy to implement, appropriate and accurate in solving the proposed model.

An efficient wavelet-based approximation method for the coupled nonlinear fractal–fractional 2D Schrödinger equations

In this study, a fast semi-discrete approach based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions of a new class of the coupled nonlinear time fractal–fractional Schrodinger equations. Although the proposed method can be applied for any fractional derivative, we focus on the Atangana–Reimann–Liouville derivative with Mittag–Leffler kernel due to its privileges. To carry out the method, first the fractal–fractional derivatives need to be discretized through the finite difference scheme and the $$\theta $$-weighted method to derive a recurrent algorithm. Then, the unknown solution of the intended problem is expanded in terms of the 2D LWs. Finally, by applying the differentiation operational matrices in each time step, the problem becomes reduced to a linear system of algebraic equations. In the proposed method, acceptable approximate solutions are achieved by employing only a few number of the basis functions. To illustrate the validity and accuracy of this practicable wavelet method, some numerical test examples are provided. The achieved numerical results manifest the exponential accuracy of approximate solutions of the new fractal–fractional model.

The impact of Legendre wavelet collocation method on the solutions of nonlinear system of two-dimensional integral equations

Numerical solutions of nonlinear system of two-dimensional integral equations have been rarely investigated in the literature. In this study, we suggest a numerically practical algorithm to approximate the solutions of nonlinear system of two-dimensional Volterra–Fredholm and Volterra integral equations. This scheme is based on two-dimensional Legendre wavelet to reduce these nonlinear systems of integral equations to a system of nonlinear algebraic equations. The main characteristic of this approach is high accuracy and computational efficiency of performing which are the consequences of Legendre wavelet properties. The main benefit of this basic function is their ability to detect singularities and their efficiency in dealing with non-sufficiently smooth function in comparison with Legendre polynomials and they minimize the error. The convergence analysis and error bound of the proposed Legendre wavelet method is investigated. Numerical examples confirm that the Legendre wavelet collocation method is accurate, reliable for solving nonlinear system of two-dimensional integral equations.

Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag–Leffler non-singular kernel

Abstract In this study, an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions of nonlinear variable-order (V-O) time fractional 2D reaction-diffusion equations. The V-O time fractional derivative is defined in the Caputo sense with Mittag-Leffler non-singular kernel of order α ( x , t ) ∈ ( 0 , 1 ) (known as the Atangana–Baleanu–Caputo fractional derivative). First, the V-O fractional derivative is approximated via the finite difference scheme and the theta-weighted method is utilized to derive a recursive algorithm. Then, the unknown solution of the intended problem is expanded via the 2D LWs. Finally, by applying the differentiation operational matrices in each time step, the solution of the problem is reduced to solution of a linear system of algebraic equations. In the proposed method, acceptable approximate solutions are achieved by employing only a few number of the basis functions. To illustrate the applicability, validity and accuracy of the presented wavelet method, some numerical test examples are solved. The achieved numerical results reveal that the established method is highly accurate in solving the introduced new V-O fractional model.

A computational wavelet method for variable-order fractional model of dual phase lag bioheat equation

In this study, we focus on the mathematical model of hyperthermia treatment as one the most constructive and effective procedures. Considering the sophisticated nature of involving phenomena in bioheat transfer inside a living tissue, several models with different levels of simplifications have been proposed. One of the general forms of the bioheat transfer equation which is introduced and studied in this paper for the first time, is the 2D-transient, dual phase lag (DPL), variable-order fractional energy equation. For finding the numerical solution of this general case, we propose an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs). Precisely, the variable-order fractional derivatives of the model are discretized in the first stage, and then the response of the model is expanded by the 2D LWs. Consequently, the main problem is transformed into an equivalent system of algebraic equations, which can be simply tackled. The stability of the proposed method is examined theoretically and experimentally. Also, the procedure is described for one example to examine the computational efficiency of method. The experimental results show the stability and spectral accuracy of the proposed method. According to the achieved results, increasing the fractional order from 0.1 to 1.0, leads to increment of maximum tissue temperatures by about 29% near the center of the targeted region.

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.

Legendre Wavelet Modified Petrov–Galerkin Method in Two-Dimensional Moving Boundary Problem

Abstract In this study, we developed the two-dimensional Legendre wavelet modified Petrov–Galerkin method for solving the two-dimensional moving boundary problem arising during melting of solid whose one surface is kept under most generalised boundary condition, and other two surfaces are insulated. The particular cases when surface subjected to the boundary condition of first, second and third kinds are discussed in detail. For validity of the present method, we have plotted graphs between residual (obtained from the original differential equation and its associated boundary conditions) and x-axis and found the effect of an error on moving layer thickness and y coordinate, respectively. Furthermore, we proved the convergence analysis of present method. The effect of parameters (Predvoditelev number, Kirpichev number, Biot number) on the moving layer thickness is discussed in detail. The whole analysis is presented in a dimensionless form.

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.

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.

Two-Dimensional Legendre Wavelets and their Applications to Integral Equations

The main focus of this paper is to present an effective numerical method for solving two-dimensional Fredholm integral equations, which appear in many phenomena in physics and engineering. For this purpose, a new set of two-dimensional wavelets is constructed by Legendre polynomials. The properties of the novelty wavelets are studied. The suggested method reduces a two-dimensional Fredholm integral equation to an algebraic equations system. Furthermore, to illustrate uniform convergence and accuracy of these wavelets, some theorems are proved. The method is applied on some examples to confirm its accuracy and computational efficiency.

Two-dimensional Legendre wavelet method for the numerical solutions of fuzzy integro-differential equations

In this paper, two dimensional Legendre wavelet method has been applied to solve the fuzzy integro-differential equations. The properties of Legendre wavelets are first presented. In fuzzy case, we have developed two dimensional Legendre wavelets to approximate the fuzzy integro-differential equations. The properties of Legendre wavelets are used to reduce the system of integral equations to a system of algebraic equations which can be solved by any usual numerical method. Illustrative examples have been discussed to demonstrate the validity and applicability of the present method.

Traveling wave solution of fractional KdV-Burger-Kuramoto equation describing nonlinear physical phenomena

In this paper, KdV-Burger-Kuramoto equation involving instability, dissipation, and dispersion parameters is solved numerically. The numerical solution for the fractional order KdV-Burger-Kuramoto (KBK) equation has been presented using two-dimensional Legendre wavelet method. The approximate solutions of nonlinear fractional KBK equation thus obtained by Legendre wavelet method are compared with the exact solutions. The present scheme is very simple, effective and convenient for obtaining numerical solution of the KBK equation.

Two-Dimensional Legendre Wavelets for Solving Time-Fractional Telegraph Equation

AbstractIn this paper, we develop an accurate and efficient Legendre wavelets method for numerical solution of the well known time-fractional telegraph equation. In the proposed method we have employed both of the operational matrices of fractional integration and differentiation to get numerical solution of the time-telegraph equation. The power of this manageable method is confirmed. Moreover the use of Legendre wavelet is found to be accurate, simple and fast.

Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions

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 fractional partial differential equations with Dirichlet boundary conditions. The proposed method is very convenient for solving such boundary value problems, since the boundary conditions are taken into account automatically. Convergence of the two-dimensional Legendre wavelets expansion is investigated. Illustrative examples are included to demonstrate the validity and applicability of the presented wavelets method.

Two-dimensional Legendre wavelets for solving fractional Poisson equation with Dirichlet boundary conditions

Abstract In this paper, the two-dimensional Legendre wavelets are applied for numerical solution of the fractional Poisson equation with Dirichlet boundary conditions. In this way, a new operational matrix of fractional derivative for the Legendre wavelets is derived and then this operational matrix has been employed to obtain the numerical solution of the above-mentioned problem. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifies the problem. The convergence of the two-dimensional Legendre wavelets expansion is investigated. Also the power of this manageable method is illustrated.