Abdel-Rehim, Entsar
[VerfasserIn]
;
Rudolf Gorenflo
[MitwirkendeR];
Francesco Mainardi
[MitwirkendeR]
Modelling and Simulating of Classical and Non-Classical Diffusion Processes by Random Walks ; Modellieren und Simulieren von klassischen und nicht-klassischen Diffusionsprozessen durch Random Walks
Titel:
Modelling and Simulating of Classical and Non-Classical Diffusion Processes by Random Walks ; Modellieren und Simulieren von klassischen und nicht-klassischen Diffusionsprozessen durch Random Walks
Beteiligte:
Abdel-Rehim, Entsar
[VerfasserIn]
Erschienen:
Freie Universität Berlin: Refubium (FU Berlin), 2004
Anmerkungen:
Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen.
Beschreibung:
1\. Title 2\. Abstract 3\. List of Abbreviations 4\. Contents 5\. List of Figures 6\. Chapter 1 1 7\. Chapter 2 5 8\. Chapter 3 25 9\. Chapter 4 43 10\. Chapter 5 69 11\. Appendix A 89 12\. Appendix B 95 13\. Appendix C 101 14\. The bibliography 103 15\. Summary 113 16\. Zusammenfassung 114 18\. Acknowledgments 116 19\. Selbständigkeitserklärung 117 ; In this thesis we discuss the equation of one-dimensional space-time fractional diffusion with drift. In Chapter 3, discrete approximations to time-fractional diffusion processes, (alpha = 2) with drift towards the origin are obtained as explicit and implicit difference schemes and as a random walk models. We have simulated these random walk models and given numerical results for the discrete approximations. Then we discuss the convergence of the discrete solutions to the stationary solutions of the model. Numerical solutions are displayed for central linear drift and for cubic central drift. Furthermore we discuss in detail the relations to the classical Ehrenfest model which is described carefully in Chapter 2. In Chapter 4, we give a survey of the theory of continuous time random walk. We show how the above space-time-time-fractional diffusion equation, with F(x)=0, can be obtained from the integral equation for a continuous time random walk or from that describing a cumulative renewal process, through well-scaled limits of vanishing waiting times and jumps. We simulate the random walk models for different values of fractional orders alpha and eta. Then we use a transformation of the independent variables x and t to simulate the random walk for space-fractional diffusion with central linear drift (i. e. F(x) = -x and eta=1). The simulation shows how jumps and waiting times are somehow compressed with respect to the case of no drift. We generalize the transformation theorem to the case of non-symmetric spatial operators. Finally in Chapter 5, we give mathematical proofs for convergence of discrete solutions of space-time-fractional diffusion without and with ...