Fractional nonlinear Volterra–Fredholm integral equations involving Atangana–Baleanu fractional derivative: framelet applications

In this work, we propose a framelet method based on B-spline functions for solving nonlinear Volterra–Fredholm integro-differential equations and by involving Atangana–Baleanu fractional derivative, which can provide a reliable numerical approximation. The framelet systems are generated using the set of B-splines with high vanishing moments. We provide some numerical and graphical evidences to show the efficiency of the proposed method. The obtained numerical results of the proposed method compared with those obtained from CAS wavelets show a great agreement with the exact solution. We confirm that the method achieves accurate, efficient, and robust measurement.

CAS WAVELETS METHOD TO SOLVE REACTION-DIFFUSION SYSTEM AND COMPARE IT WITH (G.F.E) METHOD

Wavelet analysis plays a prominent role in various fields of scientific disciplines. Mainly, wavelets are very successfully used in signal analysis for waveform representation and segmentation, time-frequency analysis, and fast algorithms in the propagation equations and reaction. This research aimed to guide researchers to use Cos and Sin (CAS) to approximate the solution of the partial differential equation system. This method has been successfully applied to solve a coupled system of nonlinear Reaction-diffusion systems. It has been shown CAS wavelet method is quite capable and suited for finding exact solutions once the consistency of the method gives wider applicability where the main idea is to transform complex nonlinear partial differential equations into algebraic equation systems, which are easy to handle and find a numerical solution for them. By comparing the numerical solutions of the CAS and Galerkin finite elements methods, the answer of nonlinear Reaction-diffusion systems using the CAS wavelets for all tˆ and x values is accurate, reliable, robust, promising, and quickly arrives at the exact solution. When parameters 𝜀1 𝑎𝑛𝑑 𝜀2 are growing and with L decreasing, then the CAS method converges to steady-state solutions quickly (the less L, the more accurate the solution). It is converging towards steady-state solutions faster than and loses steps over time. Moreover, the results also show that the solution of the CAS wavelets is more reliable and faster compared to the Galerkin finite elements (G.F.E).

Application of a new hybrid method for solving singular fractional Lane–Emden-type equations in astrophysics

In this research, a hybrid numerical method combining cosine and sine (CAS) wavelets and Green’s function approach is created to acquire the arrangements of fractional Lane–Emden Problem. The suggested methodology for detecting the solution of nonlinear equations dependent on variations of germinal algorithms is applied on nonlinear fractional Lane–Emden Problem under some smooth conditions and results in an iterative scheme of nonlinear equations Because of its efficiency, this technique is applied on a large variety of equations from the boundary value problems to the optimization. This paper is extending the suggested methodology technique for fractional Lane–Emden Problem. Moreover, the feature of the present novel method is utilized to convert the problem under observance into a system of algebraic equations that can be illuminated by suitable algorithms. A rapprochement of results has likewise been obtained using the present strategy and those reported using other techniques seem to indicate the precision and computational efficiency to establish the suitability of the Green-CAS wavelet method.

Approximate Solutions for Fractional Boundary Value Problems via Green-CAS Wavelet Method

In this study, we present a novel numerical scheme for the approximate solutions of linear as well as non-linear ordinary differential equations of fractional order with boundary conditions. This method combines Cosine and Sine (CAS) wavelets together with Green function, called Green-CAS method. The method simplifies the existing CAS wavelet method and does not require conventional operational matrices of integration for certain cases. Quasilinearization technique is used to transform non-linear fractional differential equations to linear equations and then Green-CAS method is applied. Furthermore, the proposed method has also been analyzed for convergence, particularly in the context of error analysis. Sufficient conditions for the existence of unique solutions are established for the boundary value problem under consideration. Moreover, to elaborate the effectiveness and accuracy of the proposed method, results of essential numerical applications have also been documented in graphical as well as tabular form.

Two new operational methods for solving a kind of fractional volterra integral equations

In this paper, two methods based on CAS wavelets and Legendre polynomials are applied to approximate the solutions of a kind of fractional Volterra integral equations called weakly singular integral equations. The methods are compared presenting some examples.

A Wavelet Algorithm for Fourier-Bessel Transform Arising in Optics

The aim of the paper is to propose an efficient and stable algorithm that is quite accurate and fast for numerical evaluation of the Fourier-Bessel transform of order ν, ν>-1, using wavelets. The philosophy behind the proposed algorithm is to replace the part tf(t) of the integral by its wavelet decomposition obtained by using CAS wavelets thus representing Fν(p) as a Fourier-Bessel series with coefficients depending strongly on the input function tf(t). The wavelet method indicates that the approach is easy to implement and thus computationally very attractive.

Numerical Solutions of Fractional Integrodifferential Equations of Bratu Type by Using CAS Wavelets

A numerical method based on the CAS wavelets is presented for the fractional integrodifferential equations of Bratu type. The CAS wavelets operational matrix of fractional order integration is derived. A truncated CAS wavelets series together with this operational matrix is utilized to reduce the fractional integrodifferential equations to a system of algebraic equations. The solution of this system gives the approximation solution for the truncated limited2k(2M+1). The convergence and error estimation of CAS wavelets are also given. Two examples are included to demonstrate the validity and applicability of the approach.