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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 Maple
Авторы: 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.
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Рубрика: Математика /
Статус предметного указателя: Готов указатель с номерами страниц
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Издание: 1st edition
Год издания: 2004
Количество страниц: 306
Добавлена в каталог: 10.12.2009
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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 re 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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