Numerical Algorithms For The Third-kind Volterra Integral Equations With Delay

Volterra integral equation(VIEs)has been widely used in many fields,such as biology,population dynamics and control problems.Since most of the integral equations are difficult to obtain the exact solutions,the construction of numerical algorithms to obtain numerical solutions has attracted more and more attention.At present,the theory and algorithm based on the first and second kind of Volterra integral equations have been relatively perfect,but the research progress of the third kind of Volterra integral operators with more complex integral operator structures is relatively slow.The integral operator of the third kind of Volterra integral equation is noncompact and the structure is more complex,so it is particularly important to obtain numerical solutions by using high-precision algorithms.In thesis paper,we use the pseudo-spectral Galerkin method and the Müntz collocation method to solve the third kind Volterra integral equations with proportional delays and non-Smooth Solutions and obtain the convergence analysis results.The details are as follows:In Chapter 1,we introduce the research background and the development history of the Volterra integral equation with proportional delays,and research status of numerical algorithms for this type of equation.In Chapter 2,we use the spectral and pseudo-spectral Jacobi-Galerkin method to solve the third kind of Volterra integral equations with proportional delays.First,we choose a suitable smooth transformation,so that the solutions of the obtained equivalent equations have better regularity.Secondly,the Jacobi-Guass quadrature formula is used to approximate the noncompact integral operator and the weighted inner product,and the fully discrete scheme is obtained.Then we get the error estimates of the numerical scheme under the norm of L∞ and the weighted Lωγ,ξ2 through theoretical analysis.Finally,we use numerical experiments to prove that the method has spectral convergence accuracy.In Chapter 3,we use the Muntz collocation method to solve the third kind of Volterra with proportional delays.First,by selecting a suitable fractional polynomial space as the approximate solution space,the numerical solution can more accurately approximate the analytical solution of the equation.Secondly,we use the weighted fractional Jacobi-Gauss quadrature formula to approximate the integral operator and obtain a fully discrete scheme.Then,through theoretical analysis,we obtain the error estimates of the numerical format Lnder the norm of L∞ and the weighted Lωα,β,λ·2,Finally,we use numerical experiments to verify the spectral accuracy of the numerical format.