Computer Algebra Recipes

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7.3. SIMULATING SOLITON COLLISIONS 309 equation (taking ® =1), @Ã @t @Ã + Ã @x + @3Ã =0: (7.4) @x3 They used a CDA for each ¯rst derivative, approximated @ 3 Ã=@x 3 by μ 3 @ Ã @x3 =(Ãi+2;j ¡ 2 Ãi+1;j +2Ãi¡1;j ¡ Ãi¡2;j)=(2 h P 3 ); (7.5) and averaged Ã in the nonlinear term equally over the three grid points (i+1;j), (i; j), and (i ¡ 1;j). Setting r = k=h3 , the Zabusky{Kruskal algorithm is Ãi;j+1 = Ãi;j¡1 ¡ rh 2 (Ãi+1;j + Ãi;j + Ãi¡1;j)(Ãi+1;j ¡ Ãi¡1;j)=3 ¡ r (Ãi+2;j ¡ 2 Ãi+1;j +2Ãi¡1;j ¡ Ãi¡2;j); (7.6) with j =1; 2;:::: This scheme is numerically stable for r
310 CHAPTER 7. THE HUNT FOR SOLITONS Using the operator F, we add the three time-dependent solitary waves in f, and take the time derivative of f in g. > f:= add(F(X[i],c[i]),i=1..3): g:=diff(f,t): Setting the time to zero, we plot the input pro¯le over the spatial range x =0 to M, the resulting picture being shown in Figure 7.8. > t:=0: plot(f,x=0..M,thickness=2); 2.5 2 1.5 1 0.5 0 50 100 x 150 200 250 Figure 7.8: Input pro¯le for the three-solitary-wave collision simulation. Note that there is a slight overlap of the solitary-wave tails. One could place them further apart initially, but this takes more computing time. To evaluate f and g at the spatial mesh points, we use the unapply command to turn them into operators in terms of the spatial coordinate x. > f2:=unapply(f,x): g2:=unapply(g,x): Using these two operators, we calculate the input amplitudes Ui;0 ´ f(i; t =0) and Ui;1 ´ f(i; t =0)+kg(i; t =0)fori =0toM in the ¯rst and second initial conditions, ic1 and ic2 . > ic1:=seq(U(i,0)=evalf(f2(i)),i=0..M): > ic2:=seq(U(i,1)=evalf(f2(i))+k*g2(i),i=0..M): To avoid any possible unknown U values creeping into the double do loop that will be used to iterate the numerical algorithm, we \initialize" all U values to zero for i =0toM and j =2toN, i.e., for all remaining grid points. These zeros will be overwritten as the loop is executed and new U values calculated. > init:=seq(seq(U(i,j)=0,i=0..M),j=2..N): The two initial conditions and the initialization are assigned. > assign(ic1,ic2,init):
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Richard H. Enns George C. McGuire C
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PREFACE A computer algebra system (
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viii CONTENTS 2.3.1 Finite Di®eren
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x CONTENTS 8.3 The Bifurcation Diag
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2 INTRODUCTION B. Computer Algebra
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4 INTRODUCTION (a) At what angle Á
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6 INTRODUCTION The negative answer
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8 INTRODUCTION D. Maple Help We tea
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Part I THE APPETIZERS A man ceases
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14 CHAPTER 1. PHASE-PLANE PORTRAITS
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48 CHAPTER 2. PHASE-PLANE ANALYSIS
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Chapter 3 Linear ODE Models Among a
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3.1. FIRST-ORDER MODELS 111 Therate
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3.1. FIRST-ORDER MODELS 113 Pressur
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3.1. FIRST-ORDER MODELS 115 3.1.2 G
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3.1. FIRST-ORDER MODELS 117 Evil Kn
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3.2. SECOND-ORDER MODELS 119 > tf:=
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3.2. SECOND-ORDER MODELS 121 3.2.2
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3.2. SECOND-ORDER MODELS 123 Loadin
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3.2. SECOND-ORDER MODELS 125 0.4 0.
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3.2. SECOND-ORDER MODELS 127 On ent
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3.2. SECOND-ORDER MODELS 129 Using
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3.3. SPECIAL FUNCTION MODELS 131 3.
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3.3. SPECIAL FUNCTION MODELS 133 1
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3.3. SPECIAL FUNCTION MODELS 135 Th
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3.3. SPECIAL FUNCTION MODELS 137 3.
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3.3. SPECIAL FUNCTION MODELS 139 th
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3.3. SPECIAL FUNCTION MODELS 141 PR
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3.3. SPECIAL FUNCTION MODELS 143 ¡
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3.3. SPECIAL FUNCTION MODELS 145 μ
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3.3. SPECIAL FUNCTION MODELS 147 >
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150 CHAPTER 4. NONLINEAR ODE MODELS
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208 CHAPTER 5. LINEAR PDE MODELS. P
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Chapter 6 Linear PDE Models. Part 2
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6.1. WAVE EQUATION MODELS 249 > sol
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6.1. WAVE EQUATION MODELS 251 6.1.2
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6.1. WAVE EQUATION MODELS 253 The p
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6.1. WAVE EQUATION MODELS 255 Recog
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Index acoustical waveguide, 261 ade
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INDEX 363 eigenvalue, 130 Eiram Eir
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INDEX 365 Help, Full Text Search, 8
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INDEX 367 laplace, 121,267,268 lhs,
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INDEX 369 ¯sh harvesting, 100 ¯xe
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INDEX 371 quartic map, 345 Queen Di
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Computer Algebra Recipes

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