Chapter 6 Partial Differential Equations

More documents

Recommendations

Info

$Math 1351 Sample Test 1$

$Math 3350 Fall 2004 , Sample Test # 1, Name Please note that I am ...$

$Math 3350 Sec. 6, Sample Test # 1, Name 1. Solve the separable ...$

$Matlab Lesson 6 for Math 4330 and 5344 - Texas Tech University$

$Math 4330 Sec. 1, Matlab Assignment # 1, March 24, 2006 Name 1 ...$

52 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS Exercise Set 2: Classification and Canonical Forms 1. Classify the operators and find the characteristics (if any) through the point (0, 1): (a) u tt + u xt + u xx =0 (b) u tt +4u xt +4u xx =0 (c) u tt − 4u xt + u xx =0 2. Find where the operators are hyperbolic, parabolic and elliptic: (a) Lu = u tt + tu xx + xu x (b) Lu = x 2 u tt − u xx + u (c) Lu = tu tt +2u xt + xu xx + u x 3. Transform the equation to canonical form: 2u xx − 4u xy − 6u yy + u x =0. 4. Transform the equations to canonical form: (a) e 2x u xx +2e x+y u xy + e 2y u yy + u x =0. (b) x 2 u xx + y 2 u yy = 0 for x>0, y>0. 5. Transform the equations to canonical form: (a) u xx +2u xy +17u yy =0. (b) 4u xx +12u xy +9u yy − 2u x + u =0. 6. Find the characteristics of Lu = u tt − tu xx through the point (0, 1). 7. Determine where the following equation is hyperbolic or parabolic. yu xx +(x + y)u xy + xu yy =0 Where the equation is hyperbolic, show that the general solution may be written as u(x, y) = 1 ∫ f(β) dβ + g(y − x). y − x where β = y 2 − x 2 8. Classify the following equation and find the canonical form, u xx − 2u xt + u tt =0. Show that the general solution is given by u(x, t) =tf(x + t)+g(x + t). 9. Show that (1 + x 2 )u xx +(1+y 2 )u yy + xu x + yu y = 0 is elliptic and find the canonical form of the equation.
Bibliography [1] R. Dautray and J.L. Lions, Mathematical analysis and numerical methods for science and technology, [2] G. Folland, Introduction to partial differential equations, [3] A. Friedman, Generalized functions and partial differential equations, [4] A. Friedman, Partial differential equations, [5] K.E. Gustafson Partial differential equations, [6] P.R. Garabedian, Partial Differential Equations, New York, John Wiley & Sons, 1964. [7] I.M. Gel’fand and G.E. Shilov, Generalized functions, v. 1, [8] G. Hellwig, Partial differential equations, New York, Blaisdell Publ. Co. 1964. [9] L. Hörmander, Linear partial differential operators, [10] F. John, Partial differential equations, [11] J. Kevorkian, Partial differential equations, [12] P. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shocks, [13] I.G. Petrovsky, Lectures on partial differential equations, Philadelphia, W.B. Saunders Co. 1967. [14] M. Pinsky, Introduction to partial differential equations with applications, [15] W. Rudin, Functional analysis, [16] F. Treves, Basic Partial Differential Equations, New York, Academic Press, 1975. [17] F. Treves, Linear Partial Differential Equations with Constant Coefficients, New York, Gordon & Breach, 1966. 53
Page 1 and 2: Chapter 6 Partial Differential Equa
Page 3 and 4: 6.1. INTRODUCTION 3 In this case we
Page 5 and 6: 6.1. INTRODUCTION 5 Solutions to La
Page 7 and 8: 6.1. INTRODUCTION 7 t u=0 u=1/2 u=1
Page 9 and 10: 6.2. LINEAR AND QUASILINEAR EQUATIO
Page 31 and 32: 6.3. CHARACTERISTICS AND HIGHER ORD
Page 51: 6.3. CHARACTERISTICS AND HIGHER ORD

Chapter 6 Partial Differential Equations

Create successful ePaper yourself

Delete template?

Save as template?