Analytical and Numerical Solutions for Stochastic Integral and Differential Equations in Mathematical Modelling

Abstract

The theory of deterministic chaos has enjoyed during the last three decades a rapidly increasing audience of mathematicians, physicists, engineers, biologists, economists, etc. However, this type of "chaos" can be understood only as quasi-chaos in which all states of a system can be predicted and reproduced by experiments. Meanwhile, many experiments in natural sciences have brought about hard evidence of stochastic effects. The best known example is perhaps the Brownian motion where pollen submerged in a fluid experience collisions with the molecules of the fluid and thus exhibit random motions. The study of stochasticity was initiated in the early years of the 1900's. Einstein, Smoluchowsky and Langevin wrote pioneering investigations. This work was later resumed and extended by Ornstein and Uhlenbeck. This research monograph concerns analysis of discrete-time approximations for stochastic differential equations (SDEs) driven by Wiener processes. The first chapter of the book provides a theoretical basis for working with SDEs and stochastic processes. In the present dissertation, various analytical methods like Kudryashov method, Improved sub equation method, Jacobi elliptic function (JEF) expansion method, Extended auxiliary equation method have been utilized for getting analytical solutions for stochastic differential equations viz. as Wick-type stochastic Zakharov-Kuznetsov (ZK) equation, Wick-type stochastic Kudryashov-Sinelshchikov equation, Wick-type stochastic modified Boussinesq equations, Wick-type stochastic Kersten-Krasil’shchik coupled KdV-mKdV equations and Wick-type stochastic nonlinear Schrödinger equation equations have been presented by using various analytical methods. Wavelet methodologies such as Hybrid-Legendre block pulse functions, Second kind Chebyshev wavelets, Bernstein polynomials and two dimensional second kind Chebyshev wavelets have been used to solve the stochastic integral equations. Furthermore, by applying wavelet methods, the approximate solutions of the stochastic Volterra-Fredholm integral equation, stochastic mixed Volterra-Fredholm integral equation, multi-dimensional stochastic integral equations, fractional stochastic Itô-Volterra integral equation, non-linear fractional stochastic Itô-Volterra integral equation and have been discussed in the present work. Also semi-implicit Euler-Maruyama scheme and Chebyshev spectral collocation have been applied to solve stochastic Fisher Equation and stochastic Fitzhungh Nagumo equation. These equations have lot of applications in physical phenomenon. The Fisher equation is one of the reaction-diffusion equations and is widely used in the study of biological invasion and the FitzHugh –Nagumo model is one of the classical standard models in neuroscience. Also, numerical methods viz., Euler-Maruyama method, order 1.5 strong Taylor method, Split-step forward Euler-Maruyama method, derivative free Milstein method and higher order approximation scheme have been successfully employed to fractional differential stochastic point kinetics equation for obtaining mean neutron population.