A Müntz-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.

[1]  Chuanju Xu,et al.  A fractional spectral method with applications to some singular problems , 2016, Adv. Comput. Math..

[2]  Hermann Brunner,et al.  Nonpolynomial Spline Collocation for Volterra Equations with Weakly Singular Kernels , 1983 .

[3]  Ben-yu Guo,et al.  A Multistep Legendre-Gauss Spectral Collocation Method for Nonlinear Volterra Integral Equations , 2014, SIAM J. Numer. Anal..

[4]  Sean McKee,et al.  Collocation methods for second-kind Volterra integral equations with weakly singular kernels , 1994 .

[5]  Jie Shen,et al.  Generalized Jacobi functions and their applications to fractional differential equations , 2014, Math. Comput..

[6]  Tao Tang,et al.  Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel , 2010, Math. Comput..

[7]  Giuseppe Mastroianni,et al.  Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey , 2001 .

[8]  Yanping Chen,et al.  Spectral methods for weakly singular Volterra integral equations with smooth solutions , 2009, J. Comput. Appl. Math..

[9]  David L. Ragozin,et al.  Polynomial approximation on compact manifolds and homogeneous spaces , 1970 .

[10]  Martin Stynes,et al.  A spectral collocation method for a weakly singular Volterra integral equation of the second kind , 2016, Adv. Comput. Math..

[11]  P. C. Mccarthy,et al.  Generalized Legendre polynomials , 1993 .

[12]  Sigal Gottlieb,et al.  Spectral Methods , 2019, Numerical Methods for Diffusion Phenomena in Building Physics.

[13]  Chuanju Xu,et al.  Müntz Spectral Methods for the Time-Fractional Diffusion Equation , 2017, Comput. Methods Appl. Math..

[14]  Sonia Seyed Allaei,et al.  The Jacobi Collocation Method for a Class of Nonlinear Volterra Integral Equations with Weakly Singular Kernel , 2016, J. Sci. Comput..

[15]  Tang,et al.  ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS , 2008 .

[16]  Li-Lian Wang,et al.  Sharp Error Bounds for Jacobi Expansions and Gegenbauer-Gauss Quadrature of Analytic Functions , 2012, SIAM J. Numer. Anal..

[17]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[18]  Tao Tang,et al.  Numerical Solutions for Weakly Singular Volterra Integral Equations Using Chebyshev and Legendre Pseudo-Spectral Galerkin Methods , 2016, J. Sci. Comput..

[19]  Wang Zhong-qing,et al.  An $hp$-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays , 2015 .

[20]  Tao Tang,et al.  Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind , 2012 .

[21]  Jan Lang,et al.  The generalized hardy operator with kernel and variable integral limits in banach function spaces , 1999 .

[22]  Tamás Erdélyi,et al.  Müntz systems and orthogonal Müntz-Legendre polynomials , 1994 .

[23]  Jie Shen,et al.  Spectral Methods: Algorithms, Analysis and Applications , 2011 .

[24]  Ishtiaq Ali,et al.  Spectral methods for pantograph-type differential and integral equations with multiple delays , 2009 .

[25]  Jie Shen.  Generalized Jacobi Spectral-Galerkin Method for Nonlinear Volterra Integral Equations with Weakly Singular Kernels , 2015 .

[26]  Xiang Xu,et al.  Accuracy Enhancement Using Spectral Postprocessing for Differential Equations and Integral Equations , 2008 .

[27]  David L. Ragozin,et al.  Constructive polynomial approximation on spheres and projective spaces. , 1971 .

[28]  Lars-Erik Persson,et al.  Weighted Inequalities of Hardy Type , 2003 .

[29]  Yanping Chen,et al.  A NOTE ON JACOBI SPECTRAL-COLLOCATION METHODS FOR WEAKLY SINGULAR VOLTERRA INTEGRAL EQUATIONS WITH SMOOTH SOLUTIONS * , 2013 .

[30]  H. Brunner,et al.  A SPECTRAL METHOD FOR PANTOGRAPH-TYPE DELAY DIFFERENTIAL EQUATIONS AND ITS CONVERGENCE ANALYSIS * , 2009 .

[31]  Hermann Brunner,et al.  The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes , 1985 .