Numerical studies of Korteweg-de Vries equation with random input data

Download
2018
Üreten, Mehmet Alp
Differential equations are the primary tool to mathematically model physical phenomena in industry and natural science and to gain knowledge about its features. Deterministic differential equations does not sufficiently model physically observed phenomena since there exist naturally inevitable uncertainties in nature. Employing random variables or processes as inputs or coefficients of the differential equations yields a stochastic differential equation which can clarify unnoticed features of physical events. Korteweg-de Vries (KdV) equation with the random input data is a fundamental differential equation for modeling and describing solitary waves occurring in nature. It can be represented by employing time dependent additive randomness into its forcing or space dependent multiplicative randomness into derivative of the solution. Since analytical solution of the differential equation with the random data input does not exist, quantifying and propagating uncertainty employed on the differential equation are done by numerical approximation techniques. This thesis will focus on numerical investigation of the Korteweg-de Vries equation with random input data by employing stochastic Galerkin in probability space, local discontinuous Galerkin method in spatial dimension, and theta (weighted average) method in temporal dimension. In numerical implementations, both additive noise and multiplicative noise cases are considered by comparing with other numerical techniques such as Monte Carlo and stochastic collocation methods for the probability space and finite difference method for the spatial discretization.

Suggestions

Numerical simulation of advective Lotka-Volterra systems by discontinuous Galerkin method
Aktaş, Senem; Karasözen, Bülent; Uzunca, Murat; Department of Scientific Computing (2014)
In this thesis, we study numerically advection-diffusion-reaction equations arising from Lotka-Volterra models in river ecosystems characterized by unidirectional flow. We consider two and three species models which include competition, coexistence and extinction depending on the parameters. The one dimensional models are discretized by interior penalty discontinuous Galerkin model in space. For time discretization, fully implicit backward Euler method and semi-implicit IMEX-BDF methods are used. Numerical ...
Symmetric interior penalty Galerkin method for fractional-in-space phase-field equations
Stoll, Martin; Yücel, Hamdullah (2018-01-01)
Fractional differential equations are becoming increasingly popular as a modelling tool to describe a wide range of non-classical phenomena with spatial heterogeneities throughout the applied sciences and engineering. However, the non-local nature of the fractional operators causes essential difficulties and challenges for numerical approximations. We here investigate the numerical solution of fractional-in-space phase-field models such as Allen-Cahn and Cahn-Hilliard equations via the contour integral meth...
Stochastic delay differential equations
Aladağlı, E. Ezgi; Yolcu Okur, Yeliz; Vardar Acar, Ceren; Department of Financial Mathematics (2017)
In many areas of science like physics, ecology, biology, economics, engineering, financial mathematics etc. phenomenas do not show their effect immediately at the moment of their occurrence. Generally, they influence the future states. In order to understand the structure and quantitative behavior of such systems, stochastic delay differential equations (SDDEs) are constructed while inserting the information that are obtained from the past phenomena into the stochastic differential equations (SDEs). SDDEs b...
Continuous optimization applied in MARS for modern applications in finance, science and technology
Taylan, Pakize; Weber, Gerhard Wilhelm; Yerlikaya, Fatma (2008-05-23)
Multivariate adaptive regression spline (MARS) denotes a tool from statistics, important in classification and regression, with applicability in many areas of finance, science and technology. It is very useful in high dimensions and shows a great promise for fitting nonlinear multivariate functions. The MARS algorithm for estimating the model function consists of two subalgorithms. We propose not to use the second one (backward stepwise algorithm), but we construct a penalized residual sum of squares for a ...
Time-space fractional governing equations of one-dimensional unsteady open channel flow process: Numerical solution and exploration
Ercan, Ali; Kavvas, M. Levent (2017-07-01)
Although fractional integration and differentiation have found many applications in various fields of science, such as physics, finance, bioengineering, continuum mechanics, and hydrology, their engineering applications, especially in the field of fluid flow processes, are rather limited. In this study, a finite difference numerical approach is proposed to solve the time-space fractional governing equations of 1-dimensional unsteady/non-uniform open channel flow process. By numerical simulations, results of...
Citation Formats
M. A. Üreten, “Numerical studies of Korteweg-de Vries equation with random input data,” M.S. - Master of Science, Middle East Technical University, 2018.