Ebook: Stochastic Numerics for Mathematical Physics
- Tags: Theoretical Mathematical and Computational Physics, Computational Science and Engineering, Numerical Analysis, Probability Theory and Stochastic Processes, Numerical and Computational Physics
- Series: Scientific Computation
- Year: 2004
- Publisher: Springer-Verlag Berlin Heidelberg
- Edition: 1
- Language: English
- pdf
Stochastic differential equations have many applications in the natural sciences. Besides, the employment of probabilistic representations together with the Monte Carlo technique allows us to reduce solution of multi-dimensional problems for partial differential equations to integration of stochastic equations. This approach leads to powerful computational mathematics that is presented in the treatise. The authors propose many new special schemes, some published here for the first time. In the second part of the book they construct numerical methods for solving complicated problems for partial differential equations occurring in practical applications, both linear and nonlinear. All the methods are presented with proofs and hence founded on rigorous reasoning, thus giving the book textbook potential. An overwhelming majority of the methods are accompanied by the corresponding numerical algorithms which are ready for implementation in practice. The book addresses researchers and graduate students in numerical analysis, physics, chemistry, and engineering as well as mathematical biology and financial mathematics.
Stochastic differential equations have many applications in the natural sciences. Besides, the employment of probabilistic representations together with the Monte Carlo technique allows us to reduce solution of multi-dimensional problems for partial differential equations to integration of stochastic equations. This approach leads to powerful computational mathematics that is presented in the treatise. The authors propose many new special schemes, some published here for the first time. In the second part of the book they construct numerical methods for solving complicated problems for partial differential equations occurring in practical applications, both linear and nonlinear. All the methods are presented with proofs and hence founded on rigorous reasoning, thus giving the book textbook potential. An overwhelming majority of the methods are accompanied by the corresponding numerical algorithms which are ready for implementation in practice. The book addresses researchers and graduate students in numerical analysis, physics, chemistry, and engineering as well as mathematical biology and financial mathematics.
Stochastic differential equations have many applications in the natural sciences. Besides, the employment of probabilistic representations together with the Monte Carlo technique allows us to reduce solution of multi-dimensional problems for partial differential equations to integration of stochastic equations. This approach leads to powerful computational mathematics that is presented in the treatise. The authors propose many new special schemes, some published here for the first time. In the second part of the book they construct numerical methods for solving complicated problems for partial differential equations occurring in practical applications, both linear and nonlinear. All the methods are presented with proofs and hence founded on rigorous reasoning, thus giving the book textbook potential. An overwhelming majority of the methods are accompanied by the corresponding numerical algorithms which are ready for implementation in practice. The book addresses researchers and graduate students in numerical analysis, physics, chemistry, and engineering as well as mathematical biology and financial mathematics.
Content:
Front Matter....Pages I-XIX
Mean-square approximation for stochastic differential equations....Pages 1-82
Weak approximation for stochastic differential equations....Pages 83-170
Numerical methods for SDEs with small noise....Pages 171-210
Stochastic Hamiltonian systems and Langevin-type equations....Pages 211-282
Simulation of space and space-time bounded diffusions....Pages 283-338
Random walks for linear boundary value problems....Pages 339-406
Probabilistic approach to numerical solution of the Cauchy problem for nonlinear parabolic equations....Pages 407-459
Numerical solution of the nonlinear Dirichlet and Neumann problems based on the probabilistic approach....Pages 461-507
Application of stochastic numerics to models with stochastic resonance and to Brownian ratchets....Pages 509-540
Back Matter....Pages 541-596
Stochastic differential equations have many applications in the natural sciences. Besides, the employment of probabilistic representations together with the Monte Carlo technique allows us to reduce solution of multi-dimensional problems for partial differential equations to integration of stochastic equations. This approach leads to powerful computational mathematics that is presented in the treatise. The authors propose many new special schemes, some published here for the first time. In the second part of the book they construct numerical methods for solving complicated problems for partial differential equations occurring in practical applications, both linear and nonlinear. All the methods are presented with proofs and hence founded on rigorous reasoning, thus giving the book textbook potential. An overwhelming majority of the methods are accompanied by the corresponding numerical algorithms which are ready for implementation in practice. The book addresses researchers and graduate students in numerical analysis, physics, chemistry, and engineering as well as mathematical biology and financial mathematics.
Content:
Front Matter....Pages I-XIX
Mean-square approximation for stochastic differential equations....Pages 1-82
Weak approximation for stochastic differential equations....Pages 83-170
Numerical methods for SDEs with small noise....Pages 171-210
Stochastic Hamiltonian systems and Langevin-type equations....Pages 211-282
Simulation of space and space-time bounded diffusions....Pages 283-338
Random walks for linear boundary value problems....Pages 339-406
Probabilistic approach to numerical solution of the Cauchy problem for nonlinear parabolic equations....Pages 407-459
Numerical solution of the nonlinear Dirichlet and Neumann problems based on the probabilistic approach....Pages 461-507
Application of stochastic numerics to models with stochastic resonance and to Brownian ratchets....Pages 509-540
Back Matter....Pages 541-596
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