Wavelets Operational Matrix Of Integration Research Articles

Numerical approximation of fractional SEIR epidemic model of measles and smoking model by using Fibonacci wavelets operational matrix approach

In the present article, we have considered two essential models (The epidemic model of measles and the smoking model). Across the globe, the primary cause of health problems is smoking. Measles can be controlled in infectious populations using the mathematical model representing the direct transmission of infectious diseases. The Caputo fractional derivative operator of order α∈[0,1] is used to determine the solution of the system of fractional differential equations in both models. We employed the Fibonacci wavelets operational matrix of integration and developed the Fibonacci wavelet collocation method (FWCM) to solve a nonlinear coupled system of fractional differential equations (NCSFDEs). The proposed approach transforms the given model into a system of algebraic equations. The Newton-Raphson method is considered to extract the unknown coefficient values. Numerical outcomes are obtained to illustrate the simplicity and effectiveness of the proposed method. Graphs and tables show how consistently and effectively the developed strategy works. Mathematical software called Mathematica has been used to perform all calculations. Theorems explain the convergence of this approach.

Solution of Differential Equations by Using Legendre Wavelet Operational Matrix of Integration

Solution of Differential Equations by Using Legendre Wavelet Operational Matrix of Integration

A hybrid approach based on Legendre wavelet for numerical simulation of Helmholtz equation with complex solution

In this article, with the help of Legendre wavelets a hybrid approach is presented for the numerical solution of the Helmholtz equation which has a complex solution. The present hybrid approach is based on the Legendre Wavelet Collocation Method (LWCM). Initially, the Helmholtz equation is converted to the coupled equation with suitable transformation, later all the derivatives in the equations including boundary conditions are approximated with the help of Legendre Wavelets. Later, with the help of the Legendre wavelet operational matrix of integration, a system of linear algebraic equations are formed for obtaining the numerical solution of Helmholtz equation. The aforementioned proposed algorithm is very simple and can easily be executed in any computer-oriented language efficiently. To demonstrate the efficiency and performance of the newly established technique, the method is tested on various well-known examples from previous literature including other Legendre wavelet-based methods. The obtained results produce better accuracy and widespread use of the newly developed technique for a range of benchmark problems which have complex solutions.

A novel numerical method for solution of fractional partial differential equations involving the $ \psi $-Caputo fractional derivative

<abstract><p>In this study, the $ \psi $-Haar wavelets operational matrix of integration is derived and used to solve linear $ \psi $-fractional partial differential equations ($ \psi $-FPDEs) with the fractional derivative defined in terms of the $ \psi $-Caputo operator. We approximate the highest order fractional partial derivative of the solution of linear $ \psi $-FPDE using Haar wavelets. By combining the operational matrix and $ \psi $-fractional integration, we approximate the solution and its other $ \psi $-fractional partial derivatives. Then substituting these approximations in the given $ \psi $-FPDEs, we obtained a system of linear algebraic equations. Finally, the approximate solution is obtained by solving this system. The simplicity and effectiveness of the proposed method as a mathematical tool for solving $ \psi $-Fractional partial differential equations is one of its main advantages. The sparse nature of the operational matrices improves the ability of the proposed method to execute with less computation complexity. Numerical examples are provided to show the efficiency and effectiveness of the method.</p></abstract>

Approximation of functions by Bernoulli wavelet and its applications in solution of Volterra integral equation of second kind

In this paper, two new estimators E_{2^{k-1},0}^{(1)}(f) and E_{2^{k-1},M}^{(1)}(f) of characteristic function and an estimator E_{2^{k-1},M}^{(2)}(f) of function of Hddot{text {o}}lder’s class H^{alpha } [0,1) of order 0<alpha leqslant 1 have been established using Bernoulli wavelets. A new technique has been applied for solving Volterra integral equation of second kind using Bernoulli wavelet operational matrix of integration as well as product operational matrix. These matrices have been utilized to reduce the Volterra integral equation into a system of algebraic equations, which are easily solvable. Some examples are illustrated to show the validity and efficiency of proposed technique of this research paper.

A New Boubaker Wavelets Operational Matrix of Integration

Many fields of science and engineering have used wavelet functions. They are established from expansion of a single mother wavelet function. Boubaker wavelet functions are presented in this paper based on the important properties of Boubaker polynomials. The research goal of this article is to drive a Boubaker wavelets operation matrix of integration in general formulas. Then an approximate solution method for solving a singular initial value problem is presented using Boubaker wavelets along the obtained operational matrix of integration. The importance of this method is that it converts a singular initial value problem in order to solve algebraic examples as a system. The process is based on reducing by means of integration the original problem into integral equations using a Boubaker wavelets operation matrix of integration to predict the integral equation. Illustrative experiments are included. In addition, computational results obtained by a Boubaker wavelets operation matrix of integration are compared with the exact solutions and other existing methods.

The Bernoulli wavelets operational matrix of integration and its applications for the solution of linear and nonlinear problems in calculus of variations

In this paper, a new method for solving some classes of linear and nonlinear problems in calculus of variations is presented. The method is based upon Bernoulli wavelets. The operational matrices of integration and product of Bernoulli wavelets are calculated. These matrices are then utilized to reduce the calculus of variations problems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

A novel approach of fractional-order time delay system modeling based on Haar wavelet

In this paper, fractional-order time delay system modeling is presented using Haar wavelet operational matrix of integration. Therefore, it does not require any prior knowledge of transfer function structure or partial information about fractional differentiation order. It allows the estimation of the implicit time delay parameter together with other model parameters by utilizing new delay operational matrix of Haar wavelet based on Riemann-Liouville definition. The proposed technique reduces the complexity of identification by converting the complex fractional calculus equations into simple algebra. The efficacy of the approach is verified on various integer and non-integer (fractional) order systems in simulation. For realistic condition, proposed method is verified in the presence of noise in simulation and also demonstrated on the real-time process control temperature system.

Application of Legendre Wavelets for Solving a Class of Functional Integral Equations

In this article, an algorithm based on the Legendre wavelet for solving a special class of functional linear Volterra and Fredholm integral equations has been introduced. In this technique, numerical integration rules for discrete integral and also Legendre wavelet operational matrix of integration have benefited. Then, some numerical examples are presented to demonstrate the solubility and simplicity of the proposed method.

Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems

In this paper, new operational matrix of integration is generated by using Hermite wavelets. By aid of these matrices, Hermite wavelets operational matrix method (HWOMM) is developed for second ordered nonlinear singular initial value problems. Properties of Hermite wavelets are used to convert the differential equations into system of algebraic equations which can be efficiently solved by suitable solvers. Illustrative numerical problems are considered to demonstrate the applicability of the proposed technique and those results are comparing favourably with the exact solutions and errors.

Numerical solution of two point boundary value problems by wavelet Galerkin method

In this paper, the Legendre wavelet operational matrix of integration is used to solve two point boundary value problems, in which the coefficients of the ordinary differential equation are real valued functions whose inner product with Legendre wavelet basis functions must exist. The method and convergence analysis of the Legendre wavelet is discussed. This method is applied to solve three boundary value and two moving boundary problems. In boundary value problems, we have studied the effects of condition number, elapse time and relative error on Legendre wavelet. It has been observed that the error decreases as the number of wavelet basis function increases. The condition number of square matrix of matrix equation decreases as Legendre wavelet basis function increases. The Legendre wavelet Galerkin method provides better results in lesser time, in comparison of other methods. In case of moving boundary problems the root mean square error (RMSE) for dimensionless temperature, position of moving interface and its generalized time rate are evaluated. It has been observed that the error increases as Stefan number increases.

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.

Legendre wavelets method for solving fractional integro-differential equations

In this paper, based on the constructed Legendre wavelets operational matrix of integration of fractional order, a numerical method for solving linear and nonlinear fractional integro-differential equations is proposed. By using the operational matrix, the linear and nonlinear fractional integro-differential equations are reduced to a system of algebraic equations which are solved through known numerical algorithms. The upper bound of the error of the Legendre wavelets expansion is investigated in Theorem 5.1. Finally, four numerical examples are shown to illustrate the efficiency and accuracy of the approach.

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.

Legendre wavelet Galerkin method for solving ordinary differential equations with non-analytic solution

In this article, the Legendre wavelet operational matrix of integration is used to solve boundary ordinary differential equations with non-analytic solution. Although the standard Galerkin method using Legendre polynomials does not work well for solving ordinary differential equations in which at least one of the coefficient functions or solution function is not analytic, it is shown that the Legendre wavelet Galerkin method is very efficient and suitable for solving this kind of problems. Several numerical examples are given to illustrate the efficiency and performance of the presented method.

Techniques for solving integral and differential equations by Legendre wavelets

To solve integral and differential equations, this article presents Legendre wavelets method on subintervals. This approach consists of a nonlinear function approximated by Legendre wavelets neural network, computation for Legendre wavelets operational matrix of integration, calculating product operation of Legendre wavelet vector functions and computing integer powers operation of a function approximated by Legendre wavelets. These operators are computed on the subintervals , respectively, and these computations decrease the storage and computational complexity. Using this technique, the integral and differential equations are converted into the solution of system algebraic equations. Numerical experiments demonstrate the validity and applicability of this method.

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 solution of integro-differential equations by using CAS wavelet operational matrix of integration

The CAS wavelet operational matrix P of integration is first presented and a general procedure to generate this matrix P is given. CAS wavelet approximating method is then utilized to reduce the integro-differential equation to the algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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.