Chebyshev Wavelet Method Research Articles

A Reliable Spectral Method to Reaction-Diffusion Equations in Entrapped-Cell Photobioreactor Packed with Gel Granules Using Chebyshev Wavelets.

A mathematical model of a reaction-diffusion within an entrapped-cell photobioreactor packed with gel granules containing immobilized photosynthetic bacterial cells is discussed. A theoretical model is based on a system of coupled nonlinear reaction-diffusion equations. In this research work, we have developed an efficient wavelet-based spectral approach for solving the proposed model. Analytical expressions for the concentration of substrate and product are established for all values of reaction-diffusion parameters using second kind Chebyshev wavelet method. The analytical results were also compared with Homotopy perturbation method (HPM) and Adomian decomposition method (ADM). Satisfactory agreement with ADM and HPM solutions is observed. Moreover, the use of Chebyshev wavelets is found to be simple, reliable, efficient, and computationally attractive.

Free vibration of non-uniform Euler–Bernoulli beam under various supporting conditions using Chebyshev wavelet collocation method

• Disadvantage of Chebyshev Wavelets Method is eliminated. • Non-uniform Euler–Bernoulli beams are investigated by CWCM. • The validity and applicability of the proposed method are demonstrated by examples. • Accuracy of the CWCM is quite good even for small number of grid points. This paper proposes operational matrix of r th integration of Chebyshev wavelets. A general procedure of this matrix is given. Operational matrix of r th integration is taken as r th power of operational matrix of first integration in literature. But, this study removes this disadvantage of Chebyshev wavelets method. Free vibration problems of non-uniform Euler–Bernoulli beam under various supporting conditions are investigated by using Chebyshev Wavelet Collocation Method. The proposed method is based on the approximation by the truncated Chebyshev wavelet series. A homogeneous system of linear algebraic equations has been obtained by using the Chebyshev collocation points. The determinant of coefficients matrix is equated to the zero for nontrivial solution of homogeneous system of linear algebraic equations. Hence, we can obtain i th natural frequencies of the beam and the coefficients of the approximate solution of Chebyshev wavelet series that satisfied differential equation and boundary conditions. Mode shapes functions corresponding to the natural frequencies can be obtained by normalizing of approximate solutions. The computed results well fit with the analytical and numerical results as in the literature. These calculations demonstrate that the accuracy of the Chebyshev wavelet collocation method is quite good even for small number of grid points.

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.

Two-Dimensional Chebyshev Wavelet Method for Camassa-Holm Equation with Riesz Fractional Derivative Describing Propagation of Shallow Water Waves

Two-Dimensional Chebyshev Wavelet Method for Camassa-Holm Equation with Riesz Fractional Derivative Describing Propagation of Shallow Water Waves

Chebyshev wavelet method to nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions

This research work addresses the numerical solutions of nonlinear fractional integro-differential equations with mixed boundary conditions, using Chebyshev wavelet method. The basic idea of this work started from the Caputo definition of fractional differential operator. The fractional derivatives are replaced by Caputo operator, and the solution is approximated by wavelet family of functions. The numerical scheme by Chebyshev wavelet method is constructed through a very simple and straightforward way. The numerical results of the current method are compared with the exact solutions of the problems, which show that the proposed method has a strong agreement with the exact solutions of the problems. The numerical solutions of the present method are also compared with steepest decent method and Adomian decomposition method. The comparison with other methods reveals that this method has the highest degree of accuracy than those methods.

Chebyshev wavelet method for numerical solutions of integro-differential form of Lane–Emden type differential equations

In this paper, Chebyshev wavelet method (CWM) has been applied to solve the second-order singular differential equations of Lane–Emden type. Firstly, the singular differential equation has been converted to Volterra integro-differential equation and then solved by the CWM. The properties of Chebyshev wavelets were first presented. The properties of Chebyshev wavelets via Gauss–Legendre rule were used to reduce the integral equations to a system of algebraic equations which can be solved numerically by Newton’s method. Convergence analysis of CWM has been discussed. Illustrative examples have been provided to demonstrate the validity and applicability of the present method.

Wavelet based spectral algorithm for nonlinear dynamical systems arising in ship dynamics

In this paper, we have applied an efficient shifted second kind Chebyshev wavelet method (S2KCWM) to vibrating dynamical models arising in mechanical systems such as vibration of circular membrane, damped spring mass system and ship oscillatory motions. To the best of our knowledge, until now there is no rigorous wavelet based solution has been reported for the vibrating dynamical models. The power of the manageable method is confirmed. The wavelet solutions are compared with numerical simulations by MATLAB. Good agreement between the solutions is presented in this paper. Some numerical examples are given to demonstrate the validity and applicability of the proposed method. Moreover the use of Chebyshev wavelets is found to be simple, efficient, flexible, convenient, less computation costs and computationally attractive.

On the Analysis and Numerical Formulation of Miscible Fluid Flow in Porous Media Using Chebyshev Wavelets Collocation Method

In this paper, the Chebyshev wavelet method, constructed from the Chebyshev polynomial of the first kind is proposed to numerically simulate the single-phase flow of fluid in a reservoir. The method was used together with the operational matrices of integration which resulted in an algebraic system of equations. The system of equation was solved for the wavelet coefficient and used to construct the solutions. The efficiency and accuracy of the method were demonstrated through error measurements. Both the root mean square and the maximum absolute error analysis used in the study were within significantly close range. The Chebyshev wavelet collocation method subsequently was observed to closely approximate the analytic solution to the single phase flow model quite well.

A novel attempt for finding comparatively accurate solution for sine‐Gordon equation comprising Riesz space fractional derivative

In this paper, a numerical procedure involving Chebyshev wavelet method has been implemented for computing the approximate solution of Riesz space fractional sine‐Gordon equation (SGE). Two‐dimensional Chebyshev wavelet method is implemented to calculate the numerical solution of space fractional SGE. The fractional SGE is considered as an interpolation between the classical SGE (corresponding to α = 2) and nonlocal SGE (corresponding to α = 1). As a consequence, the approximate solutions of fractional SGE obtained by using Chebyshev wavelet approach were compared with those derived by using modified homotopy analysis method with Fourier transform. Copyright © 2015 John Wiley & Sons, Ltd.

An efficient Chebyshev wavelet based analytical algorithm to steady state reaction–diffusion models arising in mathematical chemistry

A mathematical model of amperometric biosensors has been developed. The model is based on non-stationary diffusion equation containing a nonlinear term related to non-Michaelis–Menten kinetics of the reaction. In this paper, an efficient Chebyshev wavelet based approximation method is introduced for solving the steady state reaction–diffusion equations. Illustrative examples are given to demonstrate the validity and applicability of the method. The power of the manageable method is confirmed. Moreover the use of Chebyshev wavelet method is found to be simple, flexible, efficient, small computation costs and computationally attractive.

Shifted Chebyshev wavelet-quasilinearization technique for MHD squeezing flow between two infinite plates and Jeffery–Hamel flows

In this article, shifted Chebyshev wavelets method is merged with quasilinearization technique to tackle with the nonlinearity of physical problems. The accuracy of the proposed method is verified by the help of two nonlinear physical models, one MHD squeezing flow between two infinite plates and other Jeffery–Hamel flow that is obtained using proper similarity transforms. Numerical solution is also sought using Runge–Kutta order 4 method. Results obtained for different iterations and different values of degree of polynomial are described in tables and graphs which verify the accuracy and stability of the proposed method.

Modified Chebyshev wavelet methods for fractional delay-type equations

In this article, we develop the Chebyshev wavelet method for solving the fractional delay differential equations and integro-differential equations. According to the development, we approximate the delay unknown functions by the Chebyshev wavelets series at delay time, which we call the delay Chebyshev wavelet series. We also proposed a technique by combining the method of steps and Chebyshev wavelet method for solving fractional delay differential equations. This technique converts the fractional delay differential equation on a given interval to a fractional non-delay differential equation over that interval, by using the function defined on previous interval, and then apply the Chebyshev wavelet method on the obtained fractional non-delay differential equation to find the solution. Numerical examples will be presented to demonstrate the benefits of computing with these approaches.

Numerical treatment for the solution of fractional fifth-order Sawada–Kotera equation using second kind Chebyshev wavelet method

In this paper, a new method based on the Chebyshev wavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is proposed to solve time-fractional fifth-order Sawada–Kotera (SK) equation. Two-dimensional Chebyshev wavelet method is applied to compute the numerical solution of nonlinear time-fractional Sawada–Kotera equation. The approximate solutions of nonlinear time fractional Sawada–Kotera equation thus obtained by Chebyshev 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 fractional Sawada–Kotera equation.

Numerical treatment for investigation of squeezing unsteady nanofluid flow between two parallel plates

In this paper, a new method based on the Chebyshev wavelet expansion is proposed for solving a coupled system of nonlinear ordinary differential equations to model the unsteady flow of a nanofluid squeezing between two parallel plates. Chebyshev wavelet method is applied to compute the numerical solution of coupled system of nonlinear ordinary differential equations in order to model squeezing unsteady nanofluid flow. The approximate solutions of nonlinear ordinary differential equations thus obtained by Chebyshev wavelet method are compared with those of obtained by Adomian decomposition method (ADM), fourth order Runge–Kutta method and homotopy analysis method (HAM). The results obtained by the above methods are illustrated graphically and are discussed in details. The present scheme is very simple, effective and appropriate for obtaining numerical solution of squeezing unsteady nanofluid flow between parallel plates.

An Efficient Wavelet Based Approximation Method to Steady State Reaction–Diffusion Model Arising in Mathematical Chemistry

The mathematical model of Rahamathunissa and Rajendran (J Math Chem 44:849-861, 2008) in an amperometric biosensor response is discussed. In this paper, we have applied the shifted second kind Chebyshev wavelets (CW) to obtain the numerical solutions of reaction-diffusion equations containing a nonlinear term related to Michaelis-Menton kinetics of the enzymatic reaction. The application of the shifted second kind CW operational matrices for solving initial and boundary value problems is presented. The obtained numerical results demonstrate efficient and applicability of the proposed method. The power of the manageable method is confirmed. Moreover the use of shifted second kind CW method is found to be simple, efficient, accurate, small computation cost, and computationally attractive.

Chebyshev wavelets method for boundary value problems

Chebyshev wavelets method (CWM) is applied to find numerical solutions of fifth and sixth order boundary value problems. Computational work is fully supportive of compatibility of proposed algorithm and hence the same may be extended to other physical problems also. A very high level of accuracy explicitly reflects the reliability of this scheme for such problems. Key words: Chebyshev wavelets method (CWM), boundary value problems, linear and nonlinear problems, exact solutions.

An efficient wavelet based approximation method to water quality assessment model in a uniform channel

In this paper, we have developed an efficient shifted second kind Chebyshev wavelets based approximation method to water quality assessment model problem. The governing equation of the uniform flow model is one-dimensional convection diffusion equation (CDE) with variable coefficients. This water quality model requires the calculation of the substance dispersion given the water velocity in the channel. To the best of our knowledge, until now there is no rigorous wavelet solution has been reported for the water quality model problem. The shifted second kind Chebyshev wavelet method (CWM) for assessment of the chemical oxygen demand (COD) in a river is considered. Some numerical examples are presented to demonstrate the validity and applicability of the method.

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.

A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type

In this paper, we develop an accurate and efficient Chebyshev wavelets method for solution of partial differential equations with boundary conditions of the telegraph type. In the proposed method we have employed mutually the operational matrices of integration and differentiation to get numerical solutions of such equations. The power of this manageable method is confirmed. Moreover the use of Chebyshev wavelet is found to be accurate, simple and fast.

Chebyshev wavelets method for solving Bratu’s problem

A numerical method for one-dimensional Bratu’s problem is presented in this work. The method is based on Chebyshev wavelets approximates. The operational matrix of derivative of Chebyshev wavelets is introduced. The matrix together with the collocation method are then utilized to transform the differential equation into a system of algebraic equations. Numerical examples are presented to verify the efficiency and accuracy of the proposed algorithm. The results reveal that the method is accurate and easy to implement.