Computer Algebra Recipes

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5.2. DIFFUSION AND LAPLACE'S EQUATION MODELS 235 Problem 5-16: Split potential Along the circumference of a circle of radius r =1,thepotentialÁ(r; μ) has the value Á =1fortheangularrange0
236 CHAPTER 5. LINEAR PDE MODELS. PART 1 5.2.4 Hugo and the Atomic Bomb If the radiance of a thousand suns were to burst forth at once in the sky, that would be like the splendor of the Mighty One. Bhagavad Gita, a philosophical dialogue that is a sacred Hindu text, found in the Mahabharata, one of the ancient Sanskrit epics (250 BC{250 AD) Hugo, who was formerly a scientist in country X, has emigrated to the New World in search of a better life. Unfortunately, he has been forced to temporarily drive taxis until he can ¯nd a job more suitable to his educational training. While waiting for his next fare, he tries to keep his mind sharp by carrying out model calculations on his laptop computer. At this particular moment, Hugo is working on the problem of the growth of the neutron density in a nuclear chain reaction. Let's eavesdrop on what Hugo is doing. Hugo knows that if uranium nuclei are bombarded with neutrons, a given nucleus may absorb a neutron, resulting in the splitting of the uranium nucleus into two parts with the release of substantial energy as well as two or three of the neutrons that were already present in the nucleus. This splitting process is called nuclear ¯ssion andisthe¯rststepinachain reaction. Whether the reaction will keep on going depends on how many of the released neutrons are available to initiate another ¯ssion process. The factor by which the number of neutrons increases between one step and the next in the chain reaction is called the multiplication factor. In a nuclear reactor, the multiplication factor is kept at unity (called the critical condition) by using boron or cadmium control rods to \soak up" excess neutrons. In this case, the chain reaction proceeds at a constant rate with a steady output of energy. If the multiplication factor is greater than unity (the supercritical condition), the chain reaction leads to a geometrically increasing number of ¯ssions in a very short time interval with the accompanying release of an enormous amount of energy. A nuclear explosion takes place | the basis of the atomic bomb. Hugo decides that he can learn more about the underlying role that the neutrons play in the chain reaction by modeling the time evolution of some speci¯ed initial neutron distribution inside a mass of ¯ssionable material. For calculational purposes, he takes the mass to be cylindrical in shape with a radius r = a, the lower face of the cylinder at z = 0 and the upper face at z = h. For simplicity, Hugo takes the neutron density N (number of neutrons per unit volume) to be independent of the angular coordinate μ, i.e., N = N(r; z; t). In the absence of any production of neutrons by ¯ssion, the neutron density would obey the linear di®usion equation. To account for the production of neutrons by ¯ssion, Hugo adds a neutron source term ¯N,where¯is a positive rate constant, to the di®usion equation, viz., @N @t = d r2N + ¯N: (5.10) In order to use the cylindrical polar form of the Laplacian, he calls up the VectorCalculus package.
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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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288 CHAPTER 7. THE HUNT FOR SOLITON
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320 CHAPTER 8. NONLINEAR DIAGNOSTIC
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356 BIBLIOGRAPHY [Dav62] H. T. Davi
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358 BIBLIOGRAPHY [Nyq28] H. Nyquist
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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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