Chebyshev Wavelet Operational Matrix Research Articles

Wavelet Approach to Estimate The solution for Two Points Boundary Value Problems

In this paper, a new algorithm based on using Chebyshev waveletoperational matrix of integration instead of operational matrix of derivativeto find the approximate solution of two points boundary value problem.Wavelet first Chebyshev with aid of collocation method employed totransform the differential equation with its boundary condition to a systemof linear algebraic equations. A very high level of accuracy explicitlyreflected by the proposed examples.

Wavelet-Picard iterative method for solving singular fractional nonlinear partial differential equations with initial and boundary conditions

The present study applies the Picard iterative method to nonlinear singular partial fractional differential equations. The Haar and second-kind Chebyshev wavelets operational matrix of fractional integration will be used to solve problems combining linearization technique with the Picard method. The singular problem will be converted to an algebraic system of equations, which can be easily solved. Numerical examples are provided to illustrate the efficiency and accuracy of the technique.

Third-Kind Chebyshev Wavelet Method for the Solution of Fractional Order Riccati Differential Equations

In this paper, we derive the numerical solutions of the various fractional-order Riccati type differential equations using the third-kind Chebyshev wavelet operational matrix of fractional order integration (C3WOMFI) method. The operational matrix of fractional order integration method converts the fractional differential equations to a system of algebraic equations. The third-kind Chebyshev wavelet method provides sparse coefficient matrices, therefore the computational load involved for this method is not as severe and also the resulting method is faster. The numerical solutions agree with the exact solutions for non-fractional orders, and also the solutions for the fractional orders approach those of the integer orders as the fractional order coefficient [Formula: see text] approaches to 1.

Chebyshev Wavelets Method for Solving Partial Differential Equations of Fractional Order

In this paper, we use the second kind Chebyshev wavelet operational matrix of fractional integration for solving fractional order linear partial differential equations.By using this method the fractional order partial differential equation is translated into Lyapunov type matrix equation and the computation effort become convenient. Two illustrative examples are provided to demonstrate the validity, simplicity and applicability of the numerical scheme based on the Chebyshev set of functions.

An efficient method based on the second kind Chebyshev wavelets for solving variable-order fractional convection diffusion equations

ABSTRACTIn this paper, a class of variable-order fractional convection diffusion equations have been solved with assistance of the second kind Chebyshev wavelets operational matrix. The operational matrix of variable-order fractional derivative is derived for the second kind Chebyshev wavelets. By implementing the second kind Chebyshev wavelets functions and also the associated operational matrix, the considered equations will be reduced to the corresponding Sylvester equation, which can be solved by some appropriate iterative solvers. Also, the convergence analysis of the proposed numerical method to the exact solutions and error estimation are given. A variety of numerical examples are considered to show the efficiency and accuracy of the presented technique.

Solving fractional partial differential equations by using the second Chebyshev wavelet operational matrix method

Fractional partial differential equations have many applications in science and engineering. However, not only the analytical solution existed for a limited number of cases, but also the numerical methods are very complicated and difficult. The aim of this paper is to present an efficient wavelet operational method based on the second Chebyshev wavelet to solve the fractional partial differential equations. We derived the convergence of the two-dimensional second Chebyshev wavelet and give the second Chebyshev wavelet operational matrix of fractional integration. Then we present a computational method based on the above results for solving a class of fractional partial differential equations. The initial equations are transformed into a Sylvester equation. Some numerical examples are given to demonstrate the simplicity, clarity and powerfulness of the new method.

Numerical Solutions for the Eighth-Order Initial and Boundary Value Problems Using the Second Kind Chebyshev Wavelets

A collocation method based on the second kind Chebyshev wavelets is proposed for the numerical solution of eighth-order two-point boundary value problems (BVPs) and initial value problems (IVPs) in ordinary differential equations. The second kind Chebyshev wavelets operational matrix of integration is derived and used to transform the problem to a system of algebraic equations. The uniform convergence analysis and error estimation for the proposed method are given. Accuracy and efficiency of the suggested method are established through comparing with the existing quintic B-spline collocation method, homotopy asymptotic method, and modified decomposition method. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literatures.

Solving Optimal Control Linear Systems by Using New Third kind Chebyshev Wavelets Operational Matrix of Derivative

In this paper, a new third kind Chebyshev wavelets operational matrix of derivative is presented, then the operational matrix of derivative is applied for solving optimal control problems using, third kind Chebyshev wavelets expansions. The proposed method consists of reducing the linear system of optimal control problem into a system of algebraic equations, by expanding the state variables, as a series in terms of third kind Chebyshev wavelets with unknown coefficients. Example to illustrate the effectiveness of the method has been presented.

Solving Optimal Control Linear Systems by Using New Third kind Chebyshev Wavelets Operational Matrix of Derivative

In this paper, a new third kind Chebyshev wavelets operational matrix of derivative is presented, then the operational matrix of derivative is applied for solving optimal control problems using, third kind Chebyshev wavelets expansions. The proposed method consists of reducing the linear system of optimal control problem into a system of algebraic equations, by expanding the state variables, as a series in terms of third kind Chebyshev wavelets with unknown coefficients. Example to illustrate the effectiveness of the method has been presented.

Second Chebyshev wavelet operational matrix of integration and its application in the calculus of variations

In this paper, we extend the application of the second Chebyshev wavelet (SCW) method to solve variational problems and establish a clear solving procedure for this kind of problems. An operational matrix of integration based on the SCW is presented and a general procedure for forming this matrix is given. The main characteristic of SCW operational matrix method is that it can transform a variational problem to a system of algebraic equations. Thus, it can simplify the variational problems solving. The proposed method is illustrated by some applications, which can show the validity of the present approach.

Wavelet analysis method for solving linear and nonlinear singular boundary value problems

In this paper, a robust and accurate algorithm for solving both linear and nonlinear singular boundary value problems is proposed. We introduce the Chebyshev wavelets operational matrix of derivative and product operation matrix. Chebyshev wavelets expansions together with operational matrix of derivative are employed to solve ordinary differential equations in which, at least, one of the coefficient functions or solution function is not analytic. Several examples are included to illustrate the efficiency and accuracy of the proposed method.

Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet

In this paper, we first construct the second kind Chebyshev wavelet. Then we present a computational method based on the second kind Chebyshev wavelet for solving a class of nonlinear Fredholm integro-differential equations of fractional order. The second kind Chebyshev wavelet operational matrix of fractional integration is derived and used to transform the equation to a system of algebraic equations. The method is illustrated by applications and the results obtained are compared with the existing ones in open literature. Moreover, comparing the methodology with the known technique shows that the present approach is more efficient and more accurate.

Solving a nonlinear fractional differential equation using Chebyshev wavelets

Chebyshev wavelet operational matrix of the fractional integration is derived and used to solve a nonlinear fractional differential equations. Some examples are included to demonstrate the validity and applicability of the technique.

The Chebyshev wavelets operational matrix of integration and product operation matrix

Operational matrices of integration and product based on Chebyshev wavelets are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. Numerical examples are given to demonstrate applicability of these matrices.

Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration

A direct method for solving integral equations whose solutions are continuous or discontinuous by using Chebyshev wavelet basis in Galerkin equations is presented. Another method for solving Volterra type integral equations which use operational matrix of integration (OMI) for Chebyshev wavelets is introduced and used to reduce this type of integral equations to a system of algebraic equations. Some quadrature formula for calculating inner products have presented which can be operated by fast Fourier transform (FFT). The numerical examples and the number of operations show the advantages of Chebyshev Wavelet Galerkin method to some other usual methods and usual Chebyshev basis.

Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration

In this paper we present operational matrix of integration (OMI) of Chebyshev wavelets basis and the product operation matrix (POM) of it. Some comparative examples are included to demonstrate the superiority of operational matrix of Chebyshev wavelets to those of Legendre wavelets.