A M\"untz-Collocation spectral method for weakly singular volterra integral equations

In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel $(x-s)^{-\mu},0<\mu<1$. First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to derive error estimates of the numerical solution in both $L^{\infty}$- and weighted $L^{2}$-norms. The main novelty of the paper is that the proposed method is highly efficient for typical solutions that VIEs usually possess. Precisely, it is proved that the exponential convergence rate can be achieved for solutions which are smooth after the variable change $x\rightarrow x^{1/\lambda}$ for a suitable real number $\lambda$. Finally a series of numerical examples are presented to demonstrate the efficiency of the method

Approximate Numerical Solutions for Linear Volterra Integral Equations Using Touchard Polynomials

في هذا البحث، نقدم طريقة متعددة حدود توشارد لحل معادلات فولتيرا التكاملية الخطية&nbsp; من النوع الثاني والنوع الاول بالإضافة الى نوع النواة الانفرادية لهذه المعادلة. الامثلة العددية هي للتحقق من كفاءة الطريقة المقدمة وتم مقارنة الحلول العددية التقريبية مع طريقة واحدة اخرى في بعض الامثلة. جميع الحسابات&nbsp; والرسوم البيانية تم تنفيذها عن طريق برنامج الماتلاب &nbsp;2018b.In this paper, Touchard polynomials (TPs) are presented for solving Linear Volterra integral equations of the second kind (LVIEs-2k) and the first kind (LVIEs-1k) besides, the singular kernel type of this equation. Illustrative examples show the efficiency of the presented method, and the approximate numerical (AN) solutions are compared with one another method in some examples. All calculations and graphs are performed by program MATLAB2018b

A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515

A fractional B-spline collocation method for the numerical solution of fractional predator-prey models

We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost

An hp hp -version spectral collocation method for fractional Volterra integro-differential equations with weakly singular kernels

We present a multi-step spectral collocation method to solve Caputo-type fractional integro-differential equations (FIDEs) involving weakly singular kernels. We reformulate the problem as the second type Volterra integral equation (VIE) with two different weakly singular kernels. Based on these integral equations, we construct a multi-step Legendre-Gauss spectral collocation scheme for the problem. The $hp$-version convergence is established rigorously. To demonstrate the effectiveness of the suggested method and the validity of the theoretical results, the results of some numerical experiments are presented

High-Order Multivariate Spectral Algorithms for High-Dimensional Nonlinear Weakly Singular Integral Equations with Delay

One of the open problems in the numerical analysis of solutions to high-dimensional nonlinear integral equations with memory kernel and proportional delay is how to preserve the high-order accuracy for nonsmooth solutions. It is well-known that the solutions to these equations display a typical weak singularity at the initial time, which causes challenges in developing high-order and efficient numerical algorithms. The key idea of the proposed approach is to adopt a smoothing transformation for the multivariate spectral collocation method to circumvent the curse of singularity at the beginning of time. Therefore, the singularity of the approximate solution can be tailored to that of the exact one, resulting in high-order spectral collocation algorithms. Moreover, we provide a framework for studying the rate of convergence of the proposed algorithm. Finally, we give a numerical test example to show that the approach can preserve the nonsmooth solution to the underlying problems. © 2022 by the authors.King Saud University, KSUM. A. Zaky and A. Aldraiweesh extend their appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia)

Numerical solution of fractional partial differential equations by spectral methods

Fractional partial differential equations (FPDEs) have become essential tool for the modeling of physical models by using spectral methods. In the last few decades, spectral methods have been developed for the solution of time and space dimensional FPDEs. There are different types of spectral methods such as collocation methods, Tau methods and Galerkin methods. This research work focuses on the collocation and Tau methods to propose an efficient operational matrix methods via Genocchi polynomials and Legendre polynomials for the solution of two and three dimensional FPDEs. Moreover, in this study, Genocchi wavelet-like basis method and Genocchi polynomials based Ritz- Galerkin method have been derived to deal with FPDEs and variable- order FPDEs. The reason behind using the Genocchi polynomials is that, it helps to generate functional expansions with less degree and small coefficients values to derive the operational matrix of derivative with less computational complexity as compared to Chebyshev and Legendre Polynomials. The results have been compared with the existing methods such as Chebyshev wavelets method, Legendre wavelets method, Adomian decomposition method, Variational iteration method, Finite difference method and Finite element method. The numerical results have revealed that the proposed methods have provided the better results as compared to existing methods due to minimum computational complexity of derived operational matrices via Genocchi polynomials. Additionally, the significance of the proposed methods has been verified by finding the error bound, which shows that the proposed methods have provided better approximation values for under consideration FPDEs