Table Of Content

NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS MA 3243 SOLUTIONS OF PROBLEMS IN LECTURE NOTES B. Neta Department of Mathematics Naval Postgraduate School Code MA/Nd Monterey, California 93943 January 22, 2003 (cid:1)c 1996 - Professor Beny Neta 1 Contents 1 Introduction and Applications 1 1.1 Basic Concepts and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Conduction of Heat in a Rod . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 A Vibrating String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Separation of Variables-Homogeneous Equations 13 2.1 Parabolic equation in one dimension . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Other Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 13 3 Fourier Series 26 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Computation of Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 Relationship to Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.6 Fourier Cosine and Sine Series . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.7 Full solution of Several Problems . . . . . . . . . . . . . . . . . . . . . . . . 47 4 PDEs in Higher Dimensions 91 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2 Heat Flow in a Rectangular Domain . . . . . . . . . . . . . . . . . . . . . . 91 4.3 Vibrations of a rectangular Membrane . . . . . . . . . . . . . . . . . . . . . 96 4.4 Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.5 Vibrating Circular Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.6 Laplace’s Equation in a Circular Cylinder . . . . . . . . . . . . . . . . . . . 114 4.7 Laplace’s equation in a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5 Separation of Variables-Nonhomogeneous Problems 131 5.1 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 131 5.2 Method of Eigenfunction Expansions . . . . . . . . . . . . . . . . . . . . . . 136 5.3 Forced Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.4 Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.4.1 Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 160 5.4.2 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . 160 5.4.3 One Dimensional Boundary Value Problems . . . . . . . . . . . . . . 167 6 Classification and Characteristics 176 6.1 Physical Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.2 Classification of Linear Second Order PDEs . . . . . . . . . . . . . . . . . . 176 6.3 Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.4 Equations with Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . 205 i 6.5 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 6.6 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 7 Method of Characteristics 230 7.1 Advection Equation (first order wave equation) . . . . . . . . . . . . . . . . 230 7.2 Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 7.2.1 The Case S = 0, c = c(u) . . . . . . . . . . . . . . . . . . . . . . . . 245 7.2.2 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 7.2.3 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 7.2.4 Fan-like Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 263 7.2.5 Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 7.3 Second Order Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.3.1 Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.3.2 Semi-infinite String . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 7.3.3 Semi-infinite String with a Free End . . . . . . . . . . . . . . . . . . 280 7.3.4 Finite String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 8 Finite Differences 291 8.1 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 8.2 Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 8.3 Irregular Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 8.4 Thomas Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 8.5 Methods for Approximating PDEs . . . . . . . . . . . . . . . . . . . . . . . . 302 8.5.1 Undetermined coefficients . . . . . . . . . . . . . . . . . . . . . . . . 302 8.5.2 Polynomial Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 8.5.3 Integral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 8.6 Eigenpairs of a Certain Tridiagonal Matrix . . . . . . . . . . . . . . . . . . . 302 9 Finite Differences 303 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 9.2 Difference Representations of PDEs . . . . . . . . . . . . . . . . . . . . . . . 303 9.3 Heat Equation in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . 310 9.3.1 Implicit method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.3.2 DuFort Frankel method . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.3.3 Crank-Nicolson method . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.3.4 Theta (θ) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.3.5 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.3.6 Unbounded Region - Coordinate Transformation . . . . . . . . . . . . 313 9.4 Two Dimensional Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . 313 9.4.1 Explicit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.4.2 Crank Nicolson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.4.3 Alternating Direction Implicit . . . . . . . . . . . . . . . . . . . . . . 313 9.4.4 Alternating Direction Implicit for Three Dimensional Problems . . . 316 9.5 Laplace’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 ii 9.5.1 Iterative solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 9.6 Vector and Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 9.7 Matrix Method for Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 9.8 Derivative Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 320 9.9 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 9.9.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 9.9.2 Euler Explicit Method . . . . . . . . . . . . . . . . . . . . . . . . . . 325 9.9.3 Upstream Differencing . . . . . . . . . . . . . . . . . . . . . . . . . . 325 9.9.4 Lax Wendroff method . . . . . . . . . . . . . . . . . . . . . . . . . . 325 9.9.5 MacCormack Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 9.10 Inviscid Burgers’ Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 9.10.1 Lax Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 9.10.2 Lax Wendroff Method . . . . . . . . . . . . . . . . . . . . . . . . . . 328 9.10.3 MacCormack Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 9.10.4 Implicit Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 9.11 Viscous Burgers’ Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 9.11.1 FTCS method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 9.11.2 Lax Wendroff method . . . . . . . . . . . . . . . . . . . . . . . . . . 335 9.11.3 MacCormack method . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 9.11.4 Time-Split MacCormack method . . . . . . . . . . . . . . . . . . . . 335 iii List of Figures 1 Graphical solution of the eigenvalue problem . . . . . . . . . . . . . . . . . . 25 2 Graph of f(x) = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Graph of its periodic extension . . . . . . . . . . . . . . . . . . . . . . . . . 27 4 Graph of f(x) = x2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5 Graph of its periodic extension . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 Graph of f(x) = ex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7 Graph of its periodic extension . . . . . . . . . . . . . . . . . . . . . . . . . 29 8 Graph of f(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 9 Graph of its periodic extension . . . . . . . . . . . . . . . . . . . . . . . . . 30 10 Graph of f(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 11 Graph of its periodic extension . . . . . . . . . . . . . . . . . . . . . . . . . 31 12 Graph of f(x) = x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 13 Graph of its periodic extension . . . . . . . . . . . . . . . . . . . . . . . . . 32 14 graph of f(x) for problem 2c . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 15 Sketch of f(x) and its periodic extension for 1a . . . . . . . . . . . . . . . . 37 16 Sketch of the odd extension and its periodic extension for 1a . . . . . . . . . 37 17 Sketch of the even extension and its periodic extension for 1a . . . . . . . . . 38 18 Sketch of f(x) and its periodic extension for 1b . . . . . . . . . . . . . . . . 38 19 Sketch of the odd extension and its periodic extension for 1b . . . . . . . . . 38 20 Sketch of the even extension and its periodic extension for 1b . . . . . . . . . 39 21 Sketch of f(x) and its periodic extension for 1c . . . . . . . . . . . . . . . . 40 22 Sketch of the odd extension and its periodic extension for 1c . . . . . . . . . 40 23 Sketch of f(x) and its periodic extension for 1d . . . . . . . . . . . . . . . . 41 24 Sketch of the odd extension and its periodic extension for 1d . . . . . . . . . 41 25 Sketch of the even extension and its periodic extension for 1d . . . . . . . . . 42 26 Sketch of the odd extension for 2 . . . . . . . . . . . . . . . . . . . . . . . . 42 27 Sketch of the periodic extension of the odd extension for 2 . . . . . . . . . . 43 28 Sketch of the Fourier sine series for 2 . . . . . . . . . . . . . . . . . . . . . . 43 29 Sketch of f(x) and its periodic extension for problem 3 . . . . . . . . . . . . 44 30 Sketch of the even extension of f(x) and its periodic extension for problem 3 45 31 Sketch of the periodic extension of the odd extension of f(x) (problem 5) . . 46 32 Sketch of domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 33 Domain fro problem 1 of 7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 34 Domain for problem 2 of 7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 35 Maple plot of characteristics for 6.2 2a . . . . . . . . . . . . . . . . . . . . . 192 36 Maple plot of characteristics for 6.2 2a . . . . . . . . . . . . . . . . . . . . . 192 37 Maple plot of characteristics for 6.2 2b . . . . . . . . . . . . . . . . . . . . . 193 38 Maple plot of characteristics for 6.2 2b . . . . . . . . . . . . . . . . . . . . . 193 39 Maple plot of characteristics for 6.2 2c . . . . . . . . . . . . . . . . . . . . . 194 40 Maple plot of characteristics for 6.2 2d . . . . . . . . . . . . . . . . . . . . . 195 41 Maple plot of characteristics for 6.2 2d . . . . . . . . . . . . . . . . . . . . . 195 42 Maple plot of characteristics for 6.2 2f . . . . . . . . . . . . . . . . . . . . . 196 iv 43 Maple plot of characteristics for 6.3 2a . . . . . . . . . . . . . . . . . . . . . 216 44 Maple plot of characteristics for 6.3 2c . . . . . . . . . . . . . . . . . . . . . 217 45 Maple plot of characteristics for 6.3 2f . . . . . . . . . . . . . . . . . . . . . 218 46 Characteristics for problem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 237 47 Solution for problem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 48 Domain and characteristics for problem 3b . . . . . . . . . . . . . . . . . . . 250 49 Characteristics for problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 267 50 Solution for 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 51 Solution for 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 52 Sketch of initial solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 53 Solution for 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 54 Solution for 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 55 Domain for problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 56 Domain for problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 57 Domain of influence for problem 3 . . . . . . . . . . . . . . . . . . . . . . . . 287 58 Domain for problem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 59 domain for problem 1 section 9.3 . . . . . . . . . . . . . . . . . . . . . . . . 310 60 domain for problem 1 section 9.4.2 . . . . . . . . . . . . . . . . . . . . . . . 313 61 Computational Grid for Problem 2 . . . . . . . . . . . . . . . . . . . . . . . 330 v CHAPTER 1 1 Introduction and Applications 1.1 Basic Concepts and Definitions Problems 1. Give the order of each of the following PDEs a. u +u = 0 xx yy b. u +u +a(x)u +logu = f(x, y) xxx xy y c. u +u +a(x)u +u2 = f(x, y) xxx xyyy xxy d. uu +u2 +eu = 0 xx yy e. u +cu = d x y 2. Show that u(x, t) = cos(x−ct) is a solution of u +cu = 0 t x 3. Which of the following PDEs is linear? quasilinear? nonlinear? If it is linear, state whether it is homogeneous or not. a. u +u −2u = x2 xx yy b. u = u xy c. uu +xu = 0 x y d. u2 +logu = 2xy x e. u −2u +u = cosx xx xy yy f. u (1+u ) = u x y xx g. (sinu )u +u = ex x x y h. 2u −4u +2u +3u = 0 xx xy yy i. u +u u −u = 0 x x y xy 4. Find the general solution of u +u = 0 xy y (Hint: Let v = u ) y 5. Show that y u = F(xy)+xG( ) x is the general solution of x2u −y2u = 0 xx yy 1 1. a. Second order b. Third order c. Fourth order d. Second order e. First order 2. u = cos(x−ct) u = −c·(−sin(x−ct)) = csin(x−ct) t u = 1·(−sin(x−ct)) = −sin(x−ct) x ⇒ u + cu = csin(x−ct) − csin(x−ct) = 0. t x 3. a. Linear, inhomogeneous b. Linear, homogeneous c. Quasilinear, homogeneous d. Nonlinear, inhomogeneous e. Linear, inhomogeneous f. Quasilinear, homogeneous g. Nonlinear, inhomogeneous h. Linear, homogeneous i. Quasilinear, homogeneous 2 4. u + u = 0 xy y Let v = u then the equation becomes y v + v = 0 x For fixed y, this is a separable ODE dv = −dx v lnv = −x + C(y) −x v = K(y)e In terms of the original variable u we have −x u = K(y)e y −x u = e q(y) + p(x) You can check your answer by substituting this solution back in the PDE. 3 5. (cid:1) (cid:2) y u = F(xy)+xG x (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) y y y u = yF(cid:1)(xy) + G + x − G(cid:1) x x x2 x (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) y y y y y y y u = y2F(cid:1)(cid:1)(xy) + − G(cid:1) − − G(cid:1)(cid:1) + G(cid:1) xx x2 x x x2 x x2 x (cid:1) (cid:2) y2 y 2 (cid:1)(cid:1) (cid:1)(cid:1) u = y F (xy) + G xx x3 x (cid:1) (cid:2) 1 y (cid:1) (cid:1) u = xF (xy) + x G y x x (cid:1) (cid:2) 1 y 2 (cid:1)(cid:1) (cid:1)(cid:1) u = x F (xy) + G yy x x (cid:3) (cid:4) (cid:1) (cid:2) y2 1 x2u − y2u = x2 y2F(cid:1)(cid:1) + G(cid:1)(cid:1) − y2 x2F(cid:1)(cid:1) + G(cid:1)(cid:1) xx yy x3 x Expanding one finds that the first and third terms cancel out and the second and last terms cancel out and thus we get zero. 4

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Solutions of problems for PDE lectures PDF

341 Pages·1996·1.198 MB·English
by  Neta B.
#Mathematics #Computational Mathematics
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