Parallel algorithm with spectral convergence for nonlinear integro-differential equations

We discuss a numerical algorithm for solving nonlinear integro-differential equations, and illustrate our findings for the particular case of Volterra type equations. The algorithm combines a perturbation approach meant to render a linearized version of the problem and a spectral method where unknown functions are expanded in terms of Chebyshev polynomials (El-gendi's method). This approach is shown to be suitable for the calculation of two-point Green functions required in next to leading order studies of time-dependent quantum field theory.Comment: 15 pages, 9 figure

Hybrid functions approach to solve a class of Fredholm and Volterra integro-differential equations

In this paper, we use a numerical method that involves hybrid and block-pulse functions to approximate solutions of systems of a class of Fredholm and Volterra integro-differential equations. The key point is to derive a new approximation for the derivatives of the solutions and then reduce the integro-differential equation to a system of algebraic equations that can be solved using classical methods. Some numerical examples are dedicated for showing efficiency and validity of the method that we introduce

Solution of a singular integral equation by a split-interval method

The article is available at http://www.math.ualberta.ca/ijnam/Volume-4-2007/No-1-07/2007-01-05.pdf. This article is not available through the Chester Digital RepositoryThis article discusses a new numerical method for the solution of a singular integral equation of Volterra type that has an infinite class of solutions. The split-interval method is discussed and examples demonstrate its effectiveness

Improved Iterative Solution of Linear Fredholm Integral Equations of Second Kind via Inverse-Free Iterative Schemes

[EN] This work is devoted to Fredholm integral equations of second kind with non-separable kernels. Our strategy is to approximate the non-separable kernel by using an adequate Taylor's development. Then, we adapt an already known technique used for separable kernels to our case. First, we study the local convergence of the proposed iterative scheme, so we obtain a ball of starting points around the solution. Then, we complete the theoretical study with the semilocal convergence analysis, that allow us to obtain the domain of existence for the solution in terms of the starting point. In this case, the existence of a solution is deduced. Finally, we illustrate this study with some numerical experiments.This research was partially supported by a grant of the Spanish Ministerio de Ciencia, Innovacion y Universidades (Ref. PGC2018-095896-B-C21-C22).Gutiérrez, JM.; Hernández-Verón, MÁ.; Martínez Molada, E. (2020). Improved Iterative Solution of Linear Fredholm Integral Equations of Second Kind via Inverse-Free Iterative Schemes. Mathematics. 8(10):1-13. https://doi.org/10.3390/math8101747S113810Argyros, I. K. (1988). On a class of nonlinear integral equations arising in neutron transport. Aequationes Mathematicae, 36(1), 99-111. doi:10.1007/bf01837974Bruns, D. D., & Bailey, J. E. (1977). Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chemical Engineering Science, 32(3), 257-264. doi:10.1016/0009-2509(77)80203-0GANESH, M., & JOSHI, M. C. (1991). Numerical Solvability of Hammerstein Integral Equations of Mixed Type. IMA Journal of Numerical Analysis, 11(1), 21-31. doi:10.1093/imanum/11.1.21Anderson, B. D. O., & Kailath, T. (1971). Some Integral Equations with Nonsymmetric Separable Kernels. SIAM Journal on Applied Mathematics, 20(4), 659-669. doi:10.1137/0120065Ezquerro, J. A., & Hernández, M. A. (2004). A modification of the convergence conditions for Picard’s iteration. Computational & Applied Mathematics, 23(1). doi:10.1590/s0101-82052004000100003Amat, S., Ezquerro, J. A., & Hernández-Verón, M. A. (2013). Approximation of inverse operators by a new family of high-order iterative methods. Numerical Linear Algebra with Applications, 21(5), 629-644. doi:10.1002/nla.1917Barikbin, M. S., Vahidi, A. R., Damercheli, T., & Babolian, E. (2020). An iterative shifted Chebyshev method for nonlinear stochastic Itô–Volterra integral equations. Journal of Computational and Applied Mathematics, 378, 112912. doi:10.1016/j.cam.2020.112912Rabbani, M., Das, A., Hazarika, B., & Arab, R. (2020). Existence of solution for two dimensional nonlinear fractional integral equation by measure of noncompactness and iterative algorithm to solve it. Journal of Computational and Applied Mathematics, 370, 112654. doi:10.1016/j.cam.2019.11265

Joint densities of first hitting times of a diffusion process through two time dependent boundaries

Consider a one dimensional diffusion process on the diffusion interval $I$ originated in $x_0\in I$. Let $a(t)$ and $b(t)$ be two continuous functions of $t$, $t>t_0$ with bounded derivatives and with $a(t)<b(t)$ and $a(t),b(t)\in I$, $\forall t>t_0$. We study the joint distribution of the two random variables $T_a$ and $T_b$, first hitting times of the diffusion process through the two boundaries $a(t)$ and $b(t)$, respectively. We express the joint distribution of $T_a, T_b$ in terms of $P(T_a<t,T_a<T_b)$ and $P(T_bT_b)$ and we determine a system of integral equations verified by these last probabilities. We propose a numerical algorithm to solve this system and we prove its convergence properties. Examples and modeling motivation for this study are also discussed