Dissertation > Excellent graduate degree dissertation topics show

Convergence and Stability of Numerical Solution of Stochastic Differential Equations with Piecewise Continuous Arguments

Author: DaiHongYu
Tutor: LiuMingZhu
School: Harbin Institute of Technology
Course: Computational Mathematics
Keywords: Differential equations with piecewise continuous arguments Conver-gence Mean square asymptotically stable Numerical method
CLC: O211.63
Type: Master's thesis
Year: 2008
Downloads: 91
Quote: 0
Read: Download Dissertation


Differential equations with piecewise continuous arguments(EPCA) as impor-tant models are applied widely in many fields, such as physical, biological systemsand control theory. This equation had attracted much attention, and many useful con-clusion was obtained. However, up to now there is no one who has considered theinfluence of noise. Actually, the environment and accidental events may greatly in-fluence the systems. Thus analyzing stochastic differential equations with piecewisecontinuous arguments(SEPCA) is an interesting topics both in theory and applications.The paper focus on the stability of the analytical solution, convergence and sta-bility of numerical solutions for SEPCA.The paper surveys the recent developments of the study of stochastic differentialequations(SDEs) and EPCA from the point of existence, uniqueness, stability of theanalytical solution and the convergence and stability of numerical methods.For a special class of linear SEPCA, the sufficient and necessary condition whichguarantees the mean square asymptotic stability of the zero solution is given. TheEuler-Maruyama method is defined and the mean square asymptotic stability of thenumerical solution is discussed. Under the condition which guarantees the meansquare asymptotic stability of the zero solution, it is proved that the Euler-Maruymamanumerical solutions are mean square asymptotically stable if the stepsize satisfiessome restrictions. In addition, the condition of the restriction is given.For a general linear test equation, a sufficient condition which guarantees themean square asymptotic stability of the zero solution is obtained. The convergencein mean square sense and mean square asymptotic stability of the semi-implicit Eulermethod and Milstein method are studied. For the semi-implicit Euler method, severalimportant inequalities related to the analytical solution are proved and the local orderof the numerical solutions in mean square sense is obtained accordingly. Further,it is proved that the semi-implicit Euler method is convergent of order 12 in meansquare sense for the linear test equation by virtue of properties of the conditionalexpectation. For Milstein method, it is convergent with order 1 in mean square sense.By studying the difference equation, the conditions which guarantee the numerical methods to be mean square asymptotically stable are given. In addition, the conditionof the restriction of h are given.Moreover, the relevant numerical experiments are given in every part. On onehand, these experiments verify the results obtained in theory, on the other hand, theinfluences of the stepsize and parameters of the numerical methods on the stability areillustrated visually.

Related Dissertations

  1. Research on Data Acquisition and Processing Methods for Weapon Evaluation System,TJ06
  2. The Semilocal Convergence Properties of Super-Halley Method and Newton Method under Weak Conditions,O241.7
  3. Dilemma and Way Out of TV Media in Media Convergence Abstract,G206
  4. Fans Cultural Effects in Television Broadcasting,G223
  5. Feasibility Study on the Convergence of Community Education and Distance Education,G77
  6. The Discussed on the Zeros and Except Values of Complex Differences Functions,O174.5
  7. Research on the Improvements and Applications of Particle Swarm Optimization,TP18
  8. Convergence analysis of the regional energy consumption intensity,F206;F124
  9. Study on the Relationship between Industrial Structure Upgrading and Convergence Tendency of Financial Services Industry in the Central Region of China,F832.2
  10. Spectral Method for Solving Two Types of Delay Differential Equations,O241.8
  11. Research on Convergence of Regional Economic Growth Across Three Provinces in Northeast China,F127
  12. Compulsory Phase of the Basic Concepts of High School Chemistry Teaching and Study of the Convergence of Compulsory Education,G633.8
  13. Research on Modification and Application of Particle Swarm Optimization Algorithm Based on Control Methods,TP301.6
  14. TD-SCDMA and WLAN Network Convergence Technology Research,TN929.5
  15. Uniform Positioning Capabilities Engine Design and Implementation,TP311.52
  16. The Existence of Positive Solutions of a Class of Fourth-order Nonlinear Differential Equations,O175
  17. Strong Convergence Theorems for Variational Inequalities Problem and Some Relative Researches,O178
  18. Several Iterative methods and Strong Convergence Theorems for Equilibrium Problems,O177.2
  19. Shanghai World Expo news content and mode of fusion research,G206
  20. Media convergence Perspective Development of network television,F49

CLC: > Mathematical sciences and chemical > Mathematics > Probability Theory and Mathematical Statistics > Theory of probability ( probability theory, probability theory ) > Random process > Stochastic differential equation
© 2012 www.DissertationTopic.Net  Mobile