The linear barycentric rational quadrature method for Volterra integral equations

We introduce two direct quadrature methods based on linear rational interpolation for solving general Volterra integral equations of the second kind. The first, deduced by a direct application of linear barycentric rational quadrature given in former work, is shown to converge at the same rate as the rational quadrature rule but is costly on long integration intervals. The second, based on a composite version of this quadrature rule, loses one order of convergence but is much cheaper. Both require only a sample of the involved functions at equispaced nodes and yield an infinitely smooth solution of most classical examples with machine precision

The linear barycentric rational method for a class of delay Volterra integro-differential equations

A method for solving delay Volterra integro-differential equations is introduced. It is based on two applications of linear barycentric rational interpolation, barycentric rational quadrature and barycentric rational finite differences. Its zero–stability and convergence are studied. Numerical tests demonstrate the excellent agreement of our implementation with the predicted convergence orders

Numerical Solution of Fractional Volterra-Fredholm Integro-Differential Equation Using Lagrange Polynomials

في هذا البحث، ستراتيجيات جديدة لإيجاد الحل العددي للمعادلات الخطية الكسورية التفاضلية - التكاملية فولتيرا- فريدهولم (LFVFIDE) تم دراستها. الطرق المتبعه على ثلاث انواع من متعددات الحدود لاكرانج وهي: متعددة حدود لاكرانج الأصلية (OLP) ، متعددة حدود لاكرانج ذات الدعامة المركزية (BLP) و متعددة حدود لاكرانج المعدلة  (MLP).كما تم اقتراح خوارزمية عامة واعطاء  أمثلة لبرهنة فعالية الطرق وتنفيذها. وأخيرًا ، تم استخدام مقارنة بين الطرق المقترحة والطرق الأخرى لحل هذا النوع من المعادلات.In this study, a new technique is considered for solving linear fractional Volterra-Fredholm integro-differential equations (LFVFIDE's) with fractional derivative qualified in the Caputo sense. The method is established in three types of Lagrange polynomials (LP’s), Original Lagrange polynomial (OLP), Barycentric Lagrange polynomial (BLP), and Modified Lagrange polynomial (MLP). General Algorithm is suggested and examples are included to get the best effectiveness, and implementation of these types. Also, as special case fractional differential equation is taken to evaluate the validity of the proposed method. Finally, a comparison between the proposed method and other methods are taken to present the effectiveness of the proposal method in solving these problems

Explicit general linear methods with a large stability region for Volterra integro-differential equations

In this paper, we describe the construction of a class of methods with a large area of the stability region for solving Volterra integro-differential equations. In the structure of these methods which is based on a subclass of explicit general linear methods with and without Runge-Kutta stability property, we use an adequate quadrature rule to approximate the integral term of the equation. The free parameters of the methods are used to obtain methods with a large stability region. The efficiency of the proposed methods is verified with some numerical experiments and comparisons with other existing methods

Barycentric rational interpolation method for solving time-dependent fractional convection-diffusion equation

The time-dependent fractional convection-diffusion (TFCD) equation is solved by the barycentric rational interpolation method (BRIM). Since the fractional derivative is the nonlocal operator, we develop a spectral method to solve the TFCD equation to get the coefficient matrix as a full matrix. First, the fractional derivative of the TFCD equation is changed to a nonsingular integral from the singular kernel to a density function. Second, efficient quadrature of the new Gauss formula are constructed to simply compute it. Third, matrix equation of discrete the TFCD equation is obtained by the unknown function replaced by a barycentric rational interpolation basis function. Then, the convergence rate of BRIM is proved. Finally, a numerical example is given to illustrate our result

A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at $x = 0$ and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable

Error Analyses for Nyström Methods for Solving Fredholm Integral and Integro-Differential Equations

This thesis concerns the development and implementation of novel error analyses for ubiquitous Nyström-type methods used in approximating the solution in 1-D of both Fredholm integral- and integro-differential equations of the second-kind, (FIEs) and (FIDEs). The distinctive contribution of the present work is that it offers a new systematic procedure for predicting, to spectral accuracy, error bounds in the numerical solution of FIEs and FIDEs when the solution is, as in most practical applications, a priori unknown. The classic Legendre-based Nyström method is extended through Lagrange interpolation to admit solution of FIEs by collocation on any nodal distribution, in particular, those that are optimal for not only integration but also differentiation. This offers a coupled extension of optimal-error methods for FIEs into those for FIDEs. The so-called FIDE-Nyström method developed herein motivates yet another approach in which (demonstrably ill-conditioned) numerical differentiation is bypassed by reformulating FIDEs as hybrid Volterra-Fredholm integral equations (VFIEs). A novel approach is used to solve the resulting VFIEs that utilises Lagrange interpolation and Gaussian quadrature for the Volterra and Fredholm components respectively. All error bounds implemented for the above numerical methods are obtained from novel, often complex extensions of an established but hitherto-unimplemented theoretical Nyström-error framework. The bounds are computed using only the available computed numerical solution, making the methods of practical value in, e.g., engineering applications. For each method presented, the errors in the numerical solution converge (sometimes exponentially) to zero with N, the number of discrete collocation nodes; this rate of convergence is additionally confirmed via large-N asymptotic estimates. In many cases these bounds are spectrally accurate approximations of the true computed errors; in those cases that the bounds are not, the non-applicability of the theory can be predicted either a priori from the kernel or a posteriori from the numerical solution