346 Modeling and Computation H. Riecke, Northwestern University
10 Application to Partial Differential Equations 74
10.1 Unidirectional wave equation . . . . . . . . . . . . . . . . . . . . . . . 76
10.2 Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
10.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
11 Stochastic Differential Equations 84
11.1 Snippets of Ito Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 90
11.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
11.2.1 Strong Approximation . . . . . . . . . . . . . . . . . . . . . . . 97
11.2.2 Weak Approximation . . . . . . . . . . . . . . . . . . . . . . . . 103
11.2.3 Application of Weak Approximation: Feynman-Kac Formula . 105
12 Fluctuations and Tipping Points 108
13 Two-Point Boundary-Value Problems 111
13.1 Shooting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
13.2 Application: Optimal Control of a Mass in a Potential . . . . . . . . . 114
13.3 Minimization with Constraints: Pontryagin’s Principle . . . . . . . . 116
13.3.1 Pontryagin’s Principle using Functional Derivatives . . . . . . 122
13.4 Shooting Method for Linear Problems . . . . . . . . . . . . . . . . . . 124
13.5 Finite-Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . 127
14 Control of a Robotic Arm 129
15 Differential-Algebraic Equations 133
15.1 Example: Pendulum in Cartesian Coordinates . . . . . . . . . . . . . 134
15.2 Dealing with the Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
15.2.1 Enforcing by Substitution . . . . . . . . . . . . . . . . . . . . . 137
15.2.2 Brute-force constraint . . . . . . . . . . . . . . . . . . . . . . . 139
15.2.3 Solve the Constraints together with the Differential Equations 141
References 144
16 Additional Applications 145
16.1 Fluid Flow: Vortex Dynamics . . . . . . . . . . . . . . . . . . . . . . . 145
16.2 Ostwald Ripening and the Decay of Islands on Surfaces . . . . . . . . 148
16.3 Neuronal Action Potentials - Hodgkin-Huxley Model . . . . . . . . . . 152
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