Numerical solution of fractional Fredholm integro-differential equations by spectral method with fractional basis functions

This paper presents an efficient spectral method for solving the fractional Fredholm integro-differential equations. The non-smoothness of the solutions to such problems leads to the performance of spectral methods based on the classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low order of convergence. For this reason, the development of classic numerical methods to solve such problems becomes a challenging issue. Since the non-smooth solutions have the same asymptotic behavior with polynomials of fractional powers, therefore, fractional basis functions are the best candidate to overcome the drawbacks of the accuracy of the spectral methods. On the other hand, the fractional integration of the fractional polynomials functions is in the class of fractional polynomials and this is one of the main advantages of using the fractional basis functions. In this paper, an implicit spectral collocation method based on the fractional Chelyshkov basis functions is introduced. The framework of the method is to reduce the problem into a nonlinear system of equations utilizing the spectral collocation method along with the fractional operational integration matrix. The obtained algebraic system is solved using Newton's iterative method. Convergence analysis of the method is studied. The numerical examples show the efficiency of the method on the problems with smooth and non-smooth solutions in comparison with other existing methods

Higher Order Haar Wavelet Method for Solving Differential Equations

The study is focused on the development, adaption and evaluation of the higher order Haar wavelet method (HOHWM) for solving differential equations. Accuracy and computational complexity are two measurable key characteristics of any numerical method. The HOHWM introduced recently by authors as an improvement of the widely used Haar wavelet method (HWM) has shown excellent accuracy and convergence results in the case of all model problems studied. The practical value of the proposed HOHWM approach is that it allows reduction of the computational cost by several magnitudes as compared to HWM, depending on the mesh and the method parameter values used

Non-dyadic Haar Wavelet Algorithm for the Approximated Solution of Higher order Integro-Differential Equations

The objective of this study is to explore non-dyadic Haar wavelets for higher order integro-differential equations. In this research article, non-dyadic collocation method is introduced by using Haar wavelet for approximating the solution of higher order integrodifferential equations of Volterra and Fredholm type. The highest order derivatives in the integrodifferential equations are approximated by the finite series of non-dyadic Haar wavelet and then lower order derivatives are calculated by the process of integration. The integro-differential equations are reduced to a set of linear algebraic equations using the collocation approach. The Gauss - Jordan method is then used to solve the resulting system of equations. To demonstrate the efficiency and accuracy of the proposed method, numerous illustrative examples are given. Also, the approximated solution produced by the proposed wavelet technique have been compared with those of other approaches. The exact solution is also compared to the approximated solution and presented through tables and graphs. For various numbers of collocation points, different errors are calculated. The outcomes demonstrate the effectiveness of the Haar approach in resolving these equations

Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior

A Wavelet Collocation Method for some Fractional Models

This article presents an effective numerical approach based on the operational matrix of fractional order integration of Haar wavelets for dealing with the fractional models of the mixing and the Newton law of cooling problems. A general procedure of obtaining the fractional integration operational matrix of Haar wavelets which converts the fractional models into a system of algebraic equations is derived so that the computation is very simple and it is much effective than the conventional numerical methods. The reliability and the applicability of the current numerical technique for fractional models are examined by comparing the achieved results with the precise solutions

A novel numerical method for solving fractional diffusion-wave and nonlinear fredholm and volterra integral equations with zero absolute error

In this work, a new numerical method for the fractional diffusion-wave equation and nonlinear Fredholm and Volterra integro-differential equations is proposed. The method is based on Euler wavelet approximation and matrix inversion of an M × M collocation points. The proposed equations are presented based on Caputo fractional derivative where we reduce the resulting system to a system of algebraic equations by implementing the Gaussian quadrature discretization. The reduced system is generated via the truncated Euler wavelet expansion. Several examples with known exact solutions have been solved with zero absolute error. This method is also applied to the Fredholm and Volterra nonlinear integral equations and achieves the desired absolute error of 0. × 10−31 for all tested examples. The new numerical scheme is exceptional in terms of its novelty, efficiency and accuracy in the field of numerical approximation

Study of reactor constitutive model and analysis of nuclear reactor kinetics by fractional calculus approach

The diffusion theory model of neutron transport plays a crucial role in reactor theory since it is simple enough to allow scientific insight, and it is sufficiently realistic to study many important design problems. The neutrons are here characterized by a single energy or speed, and the model allows preliminary design estimates. The mathematical methods used to analyze such a model are the same as those applied in more sophisticated methods such as multi-group diffusion theory, and transport theory. The neutron diffusion and point kinetic equations are most vital models of nuclear engineering which are included to countless studies and applications under neutron dynamics. By the help of neutron diffusion concept, we understand the complex behavior of average neutron motion. The simplest group diffusion problems involve only, one group of neutrons, which for simplicity, are assumed to be all thermal neutrons. A more accurate procedure, particularly for thermal reactors, is to split the neutrons into two groups; in which case thermal neutrons are included in one group called the thermal or slow group and all the other are included in fast group. The neutrons within each group are lumped together and their diffusion, scattering, absorption and other interactions are described in terms of suitably average diffusion coefficients and cross-sections, which are collectively known as group constants. We have applied Variational Iteration Method and Modified Decomposition Method to obtain the analytical approximate solution of the Neutron Diffusion Equation with fixed source. The analytical methods like Homotopy Analysis Method and Adomian Decomposition Method have been used to obtain the analytical approximate solutions of neutron diffusion equation for both finite cylinders and bare hemisphere. In addition to these, the boundary conditions like zero flux as well as extrapolated boundary conditions are investigated. The explicit solution for critical radius and flux distributions are also calculated. The solution obtained in explicit form which is suitable for computer programming and other purposes such as analysis of flux distribution in a square critical reactor. The Homotopy Analysis Method is a very powerful and efficient technique which yields analytical solutions. With the help of this method we can solve many functional equations such as ordinary, partial differential equations, integral equations and so many other equations. It does not require enough memory space in computer, free from rounding off errors and discretization of space variables. By using the excellence of these methods, we obtained the solutions which have been shown graphically

Cubic hat-functions approximation for linear and nonlinear fractional integral-differential equations with weakly singular kernels

In the current study, a new numerical algorithm is presented to solve a class of nonlinear fractional integral-differential equations with weakly singular kernels. Cubic hat functions (CHFs) and their properties are introduced for the first time. A new fractional-order operational matrix of integration via CHFs is presented. Utilizing the operational matrices of CHFs, the main problem is transformed into a number of trivariate polynomial equations. Error analysis and the convergence of the proposed method are evaluated, and the convergence rate is addressed. Ultimately, three examples are provided to illustrate the precision and capabilities of this algorithm. The numerical results are presented in some tables and figures