Numerical Spline Method for Simulation of Stochastic Differential Equations systems

سليمان محمد محمود,أحمد الوسوف,علي سمير إحسان

doi:10.26389/ajsrp.l030621

Abstract

نقدم في هذا البحث طريقة شرائحية عددية مع وسيطي تجميع لحل نظم من المعادلات التفاضلية العشوائية متعددة الأبعاد. تمت محاكاة عملية وينر العشوائية متعددة الأبعاد المستمرة مع الزمن كعملية منفصلة، ثم دراسة الاستقرار العشوائي بمتوسط المربعات للطريقة عندما تُطَبقْ لحل نظم من المعادلات التفاضلية العشوائية الخطية من البعد الثاني. تبين الدراسة أن الطريقة تكون مستقرة ومتقاربة بمتوسط المربعات من المرتبة الثالثة عندما يتم تطبيقها لحل نظم من المعادلات التفاضلية العشوائية خطية وغير خطية. وقد تم اختبار فعالية الطريقة المقترحة بحل مسألتي اختبار الأولى خطية والثانية غير خطية، وتشير النتائج العددية إلى فعالية وكفاءة الطريقة الشرائحية المقترحة وأن النتائج الحاصلة جديرة بالاهتمام.

Highlights

Until recently, many studies ignored random effects models, due to the great difficulty of finding solutions to these models
Stochastic differential equations play an important and prominent role in multiple fields after the tremendous technological development in industrial and scientific applications and their wide uses in modeling random phenomena, and they occur in the system of differential equations that are affected by random noise, and we mention, for example, in the economics, population growth, physics, control science, medicine, biology, and mechanics, Etc
In many cases analytic solutions are not available for systems of stochastic differential equations, for these reasons, searchers numerical methods are developed to solve such systems [2, 3, 4], developments of Rung Kutta methods from various stages [5,9,10] discussed the numerical solutions of SDEs

Summary

Introduction

Many studies ignored random effects models, due to the great difficulty of finding solutions to these models. Researchers are interested in developing numerical methods to simulate analytical solutions with discrete solutions. In many cases analytic solutions are not available for systems of stochastic differential equations, for these reasons, searchers numerical methods are developed to solve such systems [2, 3, 4], developments of Rung Kutta methods from various stages [5,9,10] discussed the numerical solutions of SDEs. Linda et al [6] introduced a comparison of three different stochastic population models with regard to persistence time. Johnson [7] develop a high-order discontinuous Galerkin method for solving SDEs of Itô type driven by Wiener processes. Bayram et al [8] studied the Euler-Maruyama (EM) and Milstein methods, and to numerical solution is approximated using Monte Carlo simulation for each method. Al-Wassouf, Ehsaan where X (X1,t , X 2,t , , X m,t )T

Formulation of the Solution Method

M S-Stability of method for two-dimensional linear SD systems

Error Estimation for solution Method

Numerical Results

Conclusion