A Projection Method for Volterra Integral Equations in Weighted Spaces of Continuous Functions a Projection Method for Volterra in (original) (raw)
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A number of techniques that use variable transformations in numerical integration have been developed recently (cf. Sidi, Numerical Integration IV, H. Brass, G. H ammerlin (Eds.), Birkh auser, Basel, 1993, pp. 359 -373; Laurie, J. Comput. Appl. Math. 66 (1996) 337-344.). The use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule. In this paper the application of variable transformation techniques of Sidi and Laurie type to the numerical solution of second kind Volterra integral equations with continuous and weakly singular kernels is considered. Since the transformations are such that the end points of integration need not be used as mesh points, the methods introduced can be used for VIE with both continuous and weakly singular kernel in a uniform way. The methods have also the advantages of simplicity of application and of achieving high order of convergence. The application of the idea to Fredholm integral equations with continuous and weakly singular equations is also considered. Numerical results are included and they verify the expected increased order of convergence. They were obtained by using the trapezoidal formula for the evaluation of the transformed integrals.
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The commonly used graded piecewise polynomial collocation method for weakly singular Volterra integral equations may cause serious round-off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time-dependent equations to round-off errors. The singularity preserving (nonpolynomial) collocation method is known to have only local convergence. To overcome the shortcoming of these well-known methods, we introduce a hybrid collocation method for solving Volterra integral equations of the second kind with weakly singular kernels. In this hybrid method we combine a singularity preserving (nonpolynomial) collocation method used near the singular point of the derivative of the solution and a graded piecewise polynomial collocation method used for the rest of the domain. We prove the optimal order of global convergence for this method. The convergence analysis of this method is based on a singularity expansion of the exact solution of the equations. We prove that the solutions of such equations can be decomposed into two parts, with one part being a linear combination of some known singular functions which reflect the singularity of the solutions and the other part being a smooth function. A numerical example is presented to demonstrate the effectiveness of the proposed method and to compare it to the graded collocation method.