Analysis Of Collocation Solutions For Nonstandard Volterra Integral Equations

This paper will be concerned with high-order collocation method for the nonstandardVolterra integral equation (NVIE)(?)where u(t) is the unknown function and g, K are given functions.Many physical and biological phenomena can be modeled via the integrals (0.1) on timescales. For instance,(?)this equation is a population growth model in [5]. As a response to the need of variousapplications, recently such equations have received a considerable amount of attention onthe qualitative properties of solutions of equation (0.1)(see [4],[14],[19], [17],[18]). Forthe numerical aspects, just some special cases of (0.1) was considered, such as Volterra-Hammerstein integral equation(see [12],[13])(?)and the nonlinear second-kind Volterra integral equation as follows(?)The existence theorem and a more detailed analysis can be found in [16] or [10]([8] forlinear case). The collocation method applied to equation (0.4) can be found in [2],[1],[7],[3].Super-convergence of the collocation method was analyzed in [6](linear case [9]). The equation called nonstandard Volterra integro-di?erential equation below was stud-ied by Ma and Brunner (see [15]):(?)Motivated by the work in [15], and the fact that there is no work on the numericalsolution of this NVIE. The aims of this paper are as follows. Firstly we establish the exis-tence, uniqueness and regularity of the solution for (0.1). Secondly we apply the collocationmethod to approach this nonstandard Volterra integral equation and analyze the optimal orderof convergence of piecewise polynomial collocation approximations to its solution. Finally,we study super-convergence for the method.There are three main challenges for this NVIE:We can not directly apply the Picard iteration method to get the existence theorem asin [8].The right-hand side that includes u(t) is a new challenge for the collocation methodapplied in integral equations.If the super-convergence still exists for this special nonlinear case.We overcome these di?culties by using the Banach fixed point theorem in subintervals,adding terms to move u(t) to the left-hand side, and using the iterated solution to prove thesuper-convergence at the collocation points which is di?erent from [6], respectively.In this paper, we firstly gave the definition of Condition A, then we got the existenceand uniqueness of the solution of nonstandard Volterra integral equation. And we got theregularity properties of the solutions under certain conditions. At last, we applied collocationmethod to solve nonstandard Volterra integral equations, and obtained the convergence andsuperconvergence of collocation method.Firstly, we gave the Condition A as followsDefinition. The function K is said to satisfy Condition A, if(?) where D := {(t, s) : 0≤s≤t≤T}, 0 L(t, s) M0 and(?)Under Condition A, we obtained the theorem of existence and uniquenessTheorem. (Existence and Uniqueness) Assume(i) g∈C(I), K∈C(D×R×R),(ii) the function K satisfies Condition A.Then there exists a unique function u∈C(I) of the NVIE (0.1) on I.Under certain conditions we got the theorem of regularity as followsTheorem. (Regularity)Assume(i) g∈Cm(I) and K∈Cm(D×R×R) for some integer m 1;(ii) the function K satisfies Condition A.Then the solution u of (0.1) is Cm smooth.Then we applied collocation method to solve nonstandard Volterra integral equation andobtained the theorem of convergenceTheorem. (Convergence) Assume that(i) the functions g∈Cm(I) and K∈Cm(D×R×R) for some integer m 1;(ii) the function K satisfies Condition A;(iii) uh∈S m(??11)(Ih) is the collocation solution to (0.1) defined by (3.4) with h∈(0, hˉ).Then(?)where C depends on {ci} but not on h.At last, we got the theorem of local superconvergenceTheorem. (Local super-convergence on Xh) Assume that(i) the functions satisfy g∈Cm+1(I) and K∈Cm+2(D×R×R) for some integer 1 m 2;(ii) function K satisfies Condition A; (iii) uh∈S m(??11)(Ih)(h∈(0, hˉ)) is the collocation solution for (0.1), with collocationparameters {ci} satisfying the orthogonality condition(?)Then with uniform mesh the collocation solution is superconvergent on Xh , withmax(?)where C depends on the {ci} and on u(m+1)∞but not on h.