Physical Mathematics Michael P. Brenner September 2, 2010 School of Engineering and Applied Sciences, Harvard University Contents 1 Introduction 9 1.1 Computer Graphics and Mathematical Models . .9 1.2 Calculating while computing . 10 1.3 What is an analytical solution? . 11 2 Solving Unsolvable Problems: An Introductory Example 15 2.1 Pig farming and Polynomials . 15 2.2 Analytical Solutions to Polynomial Equations . 15 2.3 Dominant Balance and Approximate Solutions . 17 2.4 Nondimensionalization . 18 2.4.1 A Dominant Balance: ! 0...................... 19 2.4.2 The Second Dominant Balance : ! 1 ............... 21 2.5 Testing the theory with numerical simulations . 21 2.6 A bifurcation (phase transition) . 23 2.7 Lessons Learned . 24 3 Dimensions and Dimensional Analysis 25 3.1 Introduction . 25 3.2 Buckingham's π Theorem . 26 3.3 Examples . 27 3.3.1 The Pendulum . 27 3.3.2 Planetary orbits . 28 3.3.3 The size of atoms . 29 3.3.4 Fluid Viscosity . 29 3.3.5 Atomic Energy Scale . 30 3.3.6 Man's Size . 31 3.3.7 The Radius of the Earth . 31 3.3.8 Another Calculation of the Earth's Radius . 32 3.3.9 Taylor's Blast . 32 3.3.10 Pythagorean Theorem: . 33 3.3.11 McMahon's Rowers . 34 3.3.12 Lessons learned . 34 4 Polynomial Equations: Random and Otherwise 37 4.1 Solving Polynomial Equations . 37 4.2 Having courage: finding roots by iteration . 38 3 Contents 4.3 Random Quartic Polynomials . 41 4.3.1 Our first random polynomial . 41 4.3.2 Our second random polynomial equation . 42 4.3.3 nth Order Random Polynomials . 46 4.4 Having Courage . 46 4.4.1 A Problem of Hinch . 46 4.4.2 The Prime Number Theorem . 48 4.5 Numerical Approaches . 50 4.5.1 Newton's Method . 50 4.5.2 MATLAB Implementation . 51 4.5.3 Multidimensional Newton's Method . 53 5 Ordinary Differential Equations 55 5.1 A Very Simple Example . 55 5.1.1 A Simpler Way to See the Same Thing . 56 5.2 A Harder Problem . 56 5.3 Example 2 . 58 5.3.1 Doing a Better Job at Large x . 61 5.3.2 Something Disturbing . 65 5.4 A Nonlinear Example . 67 5.4.1 Negative Initial Conditions . 68 5.5 Higher Order Nonlinear Ordinary Differential Equations . 78 5.5.1 The First Balance . 79 5.5.2 The Second Balance . 79 5.5.3 The Third Balance . 80 5.5.4 Testing the Theory . 80 5.6 Numerical Solution of Ordinary Differential Equations . 81 5.6.1 Remarks . 85 6 \Simple" Integrals 87 6.1 What is a simple integral . 87 6.1.1 A more sensible definition . 88 6.2 A very easy simple integral . 89 6.2.1 What if you are on a desert island without an arctan table . 91 6.2.2 Back to the Integral . 92 6.3 A harder integral . 93 6.3.1 More accurate answers . 94 6.3.2 Practical Implementation in MATLAB . 97 6.4 Stirling's Formula . 98 6.4.1 An Estimate . 98 6.4.2 The Derivation . 99 6.5 Laplace's Method . 99 6.5.1 Proceeding more carefully :::.. .................... 101 6.5.2 Moving on to N!............................ 102 4 Contents 6.6 The Error Function . 103 6.6.1 Having Courage, once again . 106 6.6.2 Optimal Truncations . 107 6.7 Another example of an asymptotic series . 110 6.7.1 Large x behavior . 114 6.7.2 Asymptotic Series . 115 6.8 Final Remarks: Looking ahead . 115 6.8.1 Complex valued functions and essential singularities . 115 7 Convergence and Its Uses 117 7.1 What is Convergence? And is It Useful? . 117 7.2 Singularities in the Complex Plane . 119 7.2.1 Some Important Facts about Functions in the Complex Plane . 120 7.2.2 Multi-valuedness . 120 7.2.3 Differentiation and Integration . 121 7.2.4 Types of Singularities in the Complex Plane . 123 7.2.5 Summary . 124 7.3 Analytic Continuation . 125 7.3.1 Approximate Analytic Continuation . 126 7.4 Pade Approximants . 132 7.5 Appendix: The MATLAB Code Generating Complex Function on the Whole Complex Plane . 143 8 The Connection Problem 147 8.1 Matching different solutions to each other . 147 8.2 Connection problems in Classical Linear equations . 147 8.2.1 Bessel Functions . 148 8.2.2 A Less famous linear second order ordinary differential equation . 154 8.3 A nonlinear boundary value problem . 161 8.4 Spatially dependent dominant balances . 167 8.4.1 An Example of Carrier . 167 8.5 Matched Asymptotic Expansions . 169 8.5.1 When it works . 170 8.5.2 An Example from Bender and Orszag . 171 8.5.3 Some remarks on numerical Methods . 177 9 Introduction to Linear PDE's 181 9.1 Random walkers . 181 9.1.1 Random walk on a one-dimensional lattice . 181 9.1.2 Derivation #1 . 182 9.1.3 A final remark . 183 9.1.4 Derivation #2 . 183 9.1.5 Random walks in three dimensions . 184 9.1.6 Remark about Boundary Conditions . 185 5 9.1.7 Simulating Random Walkers . 185 9.2 Solving the Diffusion Equation . 186 9.2.1 Long-time Limit of the Diffusion Equation . 191 9.3 Disciplined Walkers . 192 9.3.1 Disciplined Walkers Moving in More complicated ways . 194 9.4 Biased Random Walkers . 195 9.5 Biased Not Boring Random Walkers . 196 9.6 Combining different classes of effects . 196 9.6.1 Combining Diffusion and Advection . 196 9.6.2 Fokker planck equations . 197 9.6.3 Combining Diffusion and Growth . 197 10 Integrals derived from solutions of Linear PDE's 201 10.1 Integral Transforms . 201 10.2 Contour Integration . 202 10.3 Asymptotics of Fourier-type integrals . 202 10.4 Rainbows and Oscillatory Integrals . 204 10.4.1 Colors . 207 10.4.2 Deflection angle of the rainbow and the Method of Stationary Phase207 10.4.3 Stationary Phase . ..