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Stavroulakis I.P., Tersian S.A. — Partial Differential Equations: An Introduction with Mathematica and Maple
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Название: Partial Differential Equations: An Introduction with Mathematica and MapleАвторы: Stavroulakis I.P., Tersian S.A. Аннотация: This introductory text for undergraduate and first year graduate courses explains the basic equations of mathematical physics and the properties of their solutions based on classical calculus and ordinary differential equations, including using the newest software. Stavroulakis (U. of Ioannina, Greece) and Tersian (U. of Rousse, Bulgaria) break the work down into first-order partial differential equations (PDEs), second-order PDEs, one-dimensional wave equation, one-dimensional diffusion equation, weak solutions, shock waves, conservation laws, the Laplace equation, Fourier series and Fourier methods for PDEs and diffusion and wave equations in higher dimensions. They include exercises and answers or hints for solving them.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: 1st editionГод издания: 2004Количество страниц: 306Добавлена в каталог: 10.12.2009Операции: Положить на полку |
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Предметный указатель
Absolutely integrable function 203 Almost-linear equation 2 46 Autonomous system 12 Bernoulli D. I 228 Bernoulli number 228 Bernoulli polynomial 228 Bernstein S.N. 257 Bernstein’s polynomial 257 Bessel equation 218 Bessel F.W. 218 Bessel function first kind 219 Bessel function second kind 220 Bessel inequality 218 Blow up 21 Boundary conditions, Dirichet 3 78 Boundary conditions, Neumann 4 80 Boundary conditions, Robin 4 Boundary value problem 3 Burgers J.M. 140 Canonical form 7 62 Cauchy A. 12 Cauchy Problem, Diffusion Equation 103 Cauchy problem, first order equation 13 Cauchy problem, wave equation 68 Characteristic curves 7 11 Characteristic form 60 Characteristic strip 33 Characteristic system 11 Characteristic triangle 70 Chladny E. 280 Chladny’s figures 280 Classification 46 Classification, second-order equations n var 59 Classification, second-order equations two var 46 Compatibility condition 84 Complex form Fourier Series 215 Conservation law 130 Conservation of energy 92 131 Conservation of mass 131 Conservation of momentum 131 Convergence, absolute 202 Convergence, mean-square 202 Convergence, pointwise 201 Convergence, uniform 201 Dirac kernel 124 Dirac P. 124 Dirichlet boundary condition 78 Dirichlet kernel 213 Dirichlet L.P.G. 78 Dirichlet principle 178 Discriminant 47 Domain 1 Domain of dependence 70 Domain of influence 70 D’Alembert formula 68 D’Alembert J. 68 Eigenfunction 231 Eigenvalue 231 Eigenvalue problem 230 Energy, kinetic 92 Energy, potential 92 Energy, thermal 101 Energy, total 92 Equation, almost-linear 2 Equation, biharmonic 2 Equation, Born — Infeld 2 58 Equation, Burgers 2 140 Equation, diffusion one-dim 97 Equation, diffusion or heat 2 Equation, eikonal 2 34 Equation, elliptic 48 Equation, Euler 247 Equation, fully-nonlinear 3 Equation, homogeneous 3 39 Equation, hyperbolic 48 Equation, inhomogeneous 3 40 Equation, Korteweg-de Vries 2 Equation, Laplace 2 169 Equation, linear 2 39 Equation, Liouville 151 Equation, Monge — Amp 2 Equation, nonlinear 2 Equation, parabolic 48 Equation, Poisson 169 Equation, quasi-linear 3 11 Equation, shock waves 2 Equation, transport 2 Equation, Tricomi 57 Equation, ultrahyperbolic 63 Equation, wave 2 67 Error function 73 110 Euler L. 131 First integral 23 First-order PDEs 3 Fourier cosine series 200 Fourier J.B.J. 199 Fourier method, diffusion equation 229 Fourier method, Laplace equation 243 Fourier method, wave equation 238 Fourier series 199 Fourier sine series 199 Fully-nonlinear equation 28 Function, biharmonic 181 Function, even 207 Function, Gamma 220 Function, harmonic 169 Function, odd 208 Function, piecewise continuous 204 Function, Robin 4 170 Function, subharmonic 181 Functionally independent functions 10 Fundamental solution 180 Gamma function 220 Gauss formula 174 Gauss K. 173 Gauss — Ostrogradskii formula 173 General solution 4 40 Gibbs J.W. 213 Gibbs phenomenon 213 Gradient catastrophe 21 144 Green G. 91 Green’s first identity 174 Green’s function 7 182 Green’s identity 173 Green’s second identity 174 Hadamard J. 70 Harmonic function 169 Harnack A. 193 Harnack’s first theorem 195 Harnack’s inequality 193 Harnack’s second theorem 195 Heaviside function 145 Hopf — Cole transformation 141 Huygens C. 274 Huygens’ principle 274 Hyperbolic system 131 Ill-posed problem 71 Improper integrals 104 Index, negative 62 Index, positive 62 Inhomogeneous 3 Inhomogeneous diffusion equation 118 Inhomogeneous wave equation 87 Initial value problem 3 Integral surface 12 Inverse mapping theorem 13 Irreducible equation 43 Jump discontinuity 204 kinetic energy 92 Kirchoff G.R. 269 Kirchoff’s formula 269 Laplace P.S. 169 Legendre A. 37 224 Legendre differential equation 224 Legendre polynomial 224 Legendre transformation 37 Linear equation 4 46 Linear operator 2 40 Liouville J. 151 Liouville’s equation 151 Liouville’s theorem 194 Maximum-minimum principle, diffusion equation 98 Maximum-minimum principle, Laplace equation 171 Mean value property 175 Method of shifting the data 235 Method of steepest descent 142 Mixed problem wave equation 84 Mollifiers 124 Mollifying kernels 124 Monge cone 29 Monge G. 29 Negative index 62 Neumann boundary condition 4 80 Neumann K.G. 80 Nodal set 280 Non-factorable equation 40 Nonlinear operator 2 Operator, linear 2 Operator, nonlinear 2 Order 1 Orthogonal system 200 Orthonormal system 217 Parseval M — A.Ch 205 Parseval’s equality 205 Periodic function 203 Point, elliptic 48 63 Point, hyperbolic 48 63 Point, parabolic 48 63 Point, ultrahyperbolic 63 Poisson formula, diffusion equation 103 Poisson formula, Laplace equation 189 190 Poisson S.D. 103 potential energy 92 Principal part 47 Quasi-linear equation 3 11 Rank 10 Rankine — Hugoniot condition 164 Reducible equation 40 Reflection method 78 Region 1 Regularizations 125 Riemann G.F. 154 Riemann problem 154 Schwarz H.A. 125 Second-order PDEs 39 Shock speed 155 Shock wave 123 155 Sobolev S.L. 124 Sobolev space 124 Solution 1 Spherical means 269 Spherical wave equation 76 Steepest descent 142 Strictly hyperbolic system 136 Strip condition 33 Sturm — Liouville problem 230 Superposition 231 Swimmer effect 83 Symmetry of Green’s function 183 Taylor B. 142 thermal conductivity 97 Thermal energy 101 Transpose matrix 61 Ultrahyperbolic equation 63 Variation of constants (param.) 234 Wave equation 67 Weak derivative 123 Weak solution 123 135 Weierstrass criterion 105 202 Weierstrass K.W. 105 Well-posedness 70 170
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