Jacobi pseudo-spectral Galerkin method for second kind Volterra integro-differential equations with a weakly singular kernel

The Jacobi pseudo-spectral Galerkin method for the Volterra integro-differential equations of the second kind with a weakly singular kernel is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the Lωα,β2-norm and the L∞-norm) will decay exponentially provided that the source function is sufficiently smooth. Numerical examples are given to illustrate the theoretical results

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

Murruliste tuletistega diferentsiaalvõrrandite ligikaudne lahendamine

Murrulised tuletised (s.t. tuletised, mille järk ei ole täisarv) on pakkunud huvi juba alates ajast, millal I. Newton ja G. W. Leibniz rajasid matemaatilise analüüsi aluseks oleva diferentsiaal- ja integraalarvutuse. Kaua aega käsitleti murruliste tuletistega seotud küsimusi vaid teoreetilisest vaatepunktist, sest ei olnud näha, millised võiksid olla murruliste tuletiste rakendusvõimalused. Viimastel aastakümnetel on aga leitud, et murrulisi tuletisi sisaldavad diferentsiaalvõrrandid kirjeldavad mitmesuguste materjalide ja protsesside käitumist paremini kui täisarvulist järku tuletistega diferentsiaalvõrrandid. Kuna murruliste tuletistega diferentsiaalvõrrandite täpse lahendi leidmine ei ole enamasti võimalik, peame nende lahendeid leidma ligikaudselt. See nõuab spetsiaalsete meetodite väljatöötamist, sest murruliste tuletistega diferentsiaalvõrrandite korral ei ole reeglina rakendatavad täisarvuliste tuletistega diferentsiaalvõrrandite vallast tuntud tulemused. Käesolevas väitekirjas uuritakse murruliste tuletistega diferentsiaalvõrrandi lahendi siledust ja saadud informatsiooni alusel töötatakse välja kõrget järku täpsusega lahendusalgoritmid niisuguste võrrandite ligikaudseks lahendamiseks. Saadud teoreetilisi tulemusi kontrollitakse arvukate numbriliste eksperimentidega mitmesugustel testvõrranditel.The concept of a fractional derivative can be traced back to the end of the seventeenth century, the time when Newton and Leibniz developed the foundations of differential and integral calculus. Despite this, for a long time, considerations regarding fractional derivatives were purely theoretical treatments for which there were no serious practical applications. It is only during the last decades that there has been a spectacular increase of studies regarding fractional derivatives and differential equations with such derivatives, mainly because of new applications of fractional derivatives in several fields of applied science. However, when working with problems stemming from real-world applications, it is only rarely possible to find the exact solution of a given fractional differential equation, and even if such an analytic solution is available, it is typically too complicated to be used in practice. Therefore numerical methods specialized for solving fractional differential equations are required. In the present thesis the regularity properties of the exact solutions of a wide class of fractional differential and integro-differential equations are investigated. Based on the obtained regularity properties, the numerical solution of the problem is discussed, the convergence of proposed algorithms is proven and their global convergence estimates are derived. The obtained theoretical results are supported by many numerical experiments with various test problems.https://www.ester.ee/record=b529526

A NUMERICAL METHOD FOR SOLVING SYSTEMS OF HYPERSINGULAR INTEGRO-DIFFERENTIAL EQUATIONS

This paper is concerned with a collocation-quadrature method for solving systems of Prandtl's integro-differential equations based on de la Vallee Poussin filtered interpolation at Chebyshev nodes. We prove stability and convergence in Holder-Zygmund spaces of locally continuous functions. Some numerical tests are presented to examine the method's efficacy

A fast algorithm for Prandtl's integro-differential equation

AbstractCollocation and quadrature methods for singular integro-differential equations of Prandtl's type are studied in weighted Sobolev spaces. A fast algorithm basing on the quadrature method is proposed. Convergence results and error estimates are given

Quadrature methods for integro-differential equations of Prandtl's type in weighted spaces of continuous functions

The paper deals with the approximate solution of integro-differential equations of Prandtl's type. Quadrature methods involving ``optimal'' Lagrange interpolation processes are proposed and conditions under which they are stable and convergent in suitable weighted spaces of continuous functions are proved. The efficiency of the method has been tested by some numerical experiments, some of them including comparisons with other numerical procedures. In particular, as an application, we have implemented the method for solving Prandtl's equation governing the circulation air flow along the contour of a plane wing profile, in the case of elliptic or rectangular wing-shape.Comment: 34 page

Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations

This article addresses the solution of multi-dimensional integro-differential equations (IDEs) by means of the spectral collocation method and taking the advantage of the properties of shifted Jacobi polynomials. The applicability and accuracy of the present technique have been examined by the given numerical examples in this paper. By means of these numerical examples, we ensure that the present technique is simple and very accurate. Furthermore, an error analysis is performed to verify the correctness and feasibility of the proposed method when solving IDE