Boundary-value Problems in Rectangular Coordinates
|
|
- Lenard Bennett
- 6 years ago
- Views:
Transcription
1 Boundary-value Problems in Rectangular Coordinates 2009
2 Outline Separation of Variables: Heat Equation on a Slab Separation of Variables: Vibrating String Separation of Variables: Laplace Equation Review on Boundary Conditions Dirichlet s Problems Neumann s Problems Robin s Problems(Optional) 2D Heat Equation 2D Wave Equation
3 Separation of Variables Separation of Variables
4 One Dimensional Heat Equation (Heat Conduction on a Slab) Problem: Consider a unifrom slab (or rod or bar) of length L with insulated lateral surface. Let the internal temperature distribution on the slab be u(x, t) at point x and time t. Given that at time t = the temperature distribution of the slab is f(x), and given that both ends on the slab are held at zero constant temperature. Find the subsequent temperature distribution u(x, t) for 0 < x < L and t > 0 on the slab. (Of course, from our experience we know that u 0 as t, and our solution for u(x, t) should also capture this observation.)
5 Boundary-value Problem 1D Heat Equation: u t = 2 u c2, x2 0 < x < L, t > 0; (1) Boundary Conditions (Zero temperature at both ends): u(0, t) = 0 and u(l, t) = 0, t > 0; (2) Initial condition (Initial temperature distribution f(x)): u(x, 0) = f(x), 0 < x < L. (3)
6 Separation of variables Let u(x, t) = X(x)T (t), differentiate & subsitute into Eq.(1): T c 2 T = X X = k, Which gives a set of two ODEs: Now, the boundary condition becomes X kx = 0, (4) T kc 2 T = 0. (5) u(0, t) = X(0)T (t) = 0 = X(0) = 0 t > 0, (6) u(l, t) = X(L)T (t) = 0 = X(L) = 0 t > 0. (7)
7 There are three possible cases for the eigenvalue k: k = b 2 > 0 X(x) = c1 cosh bx + c 2 sinh bx, X(0) = X(L) = 0 = c1 = c 2 = 0 trivial solution k = 0 X(x) = c1 x + c 2 X(0) = X(L) = 0 = c1 = c 2 = 0 trivial solution k = β 2 < 0 X(x) = c1 cos βx + c 2 sin βx X(0) = 0 = c1 = 0 X(L) = 0 = β = βn = nπ L, n = 1, 2,... c2 is arbitrary, and set c 2 to 1.
8 Thus, a non-trivial solution for Eq.(4) is X(x) = X n (x) = sin nπ x, n = 1, 2,.... L By using k = β 2, solve Eq.(5) T (t), T (t) = T n (t) = b n e λ2 nt, n = 1, 2,..., where λ n = cnπ L. Combining X n (x) and T n (t), we get a solution for Eq.(1) and the solution satisfies the Boundary Conditions Eq.(2) u n (x, t) = X n (x)t n (t) = b n e λ2 n t sin nπ L x, n = 1, 2,..., (8)
9 Principle of Superposition Theorem If φ and ψ are solutions to a linear differential equation and satisfy a linear boundary condition, then the linear combination u = c 1 φ + c 2 ψ is also a solution and satisfies the same boundary condition. Here c 1 and c 2 are constants. Since u 1 (x, t), u 2 (x, t),... are satisfying the 1D Heat equation and the zero temperature boundary conditions. Thus, a general solution is the superposition of all these u n (x, t): u(x, t) = n=1 b n e λ2 nt sin nπ L x. (9)
10 Half-range Fourier Series Applying initial condition u(x, 0) = b n sin nπ x = f(x), 0 < x < L, L = b n = 2 L k=1 L 0 f(x) sin nπ x dx, n = 1, 2,... L
11 Summary Boundary-value problems General solution u = c 2 2 u, 0 < x < L, t > 0; t t2 u(0, t) = 0 and u(l, t) = 0, t > 0; u(x, 0) = f(x), 0 < x < L. u(x, t) = b n = 2 L n=1 L 0 b n e λ2 n t sin nπ L x, f(x) sin nπx L dx. λ n = cnπ L ;
12 Sumarry for Method of Separation of Variables 1. Decompose u into products of functions of one variable. 2. Decompose the PDE into a set of ODEs. 3. Identify boundary conditions and the corresponding Sturm-Liouville problems. 4. Solve the Sturm-Liouville problems and obtain the corresponding eigenvalues. 5. Apply principle of superposition to obtain the eigenfucntion expansion of a general solution. 6. Use the initial conditions to obtain generalized Fourier coefficients of the eigenfunction expansion.
13 Example Given that a homogeneous rod of lenght L = π is properly insulated except at both ends. Suppose that both ends of the rod are kept at a constant temperature of zero degree Celsius. Find the temperature distribution in the rod u(x, t) for t > 0 if given that the entire rod is initially at a temperature of 100 degree Celsius.
14 One Dimensional Wave Equation (Vibrating String) Problem: Consider a stretched string of length L with both ends fastened on the x-axis. Suppose that the string is plucked from its equilibrium position and release at time t = 0. Assuming that the amplitude of the vibration at time t 0 and position x is u(x, t). Suppose that the initial shape of the string is u(x, 0) = f(x) and the initial velocity at each point on the string is du (x, 0) = g(x). dx Find the subsequent motion of the string u(x, t) for 0 < x < L and t > 0.
15 Boundary-value problems 1D Wave Equation: 2 u t 2 = 2 u c2, 0 < x < L, t > 0; (10) t2 Boundary Conditions (fixed end points): u(0, t) = 0 and u(l, t) = 0, t > 0; (11) Initial Conditions (initial displacement and initial velocity): u(x, 0) = f(x) and u (x, 0) = g(x), 0 < x < L. (12) t
16 Separation of variables Let u(x, t) = X(x)T (t), differentiate and subsitute into Eq. 10: T c 2 T = X X = k, Which gives a set of two ODEs: Now, the boundary condition becomes X kx = 0, (13) T kc 2 T = 0. (14) u(0, t) = X(0)T (t) = 0 = X(0) = 0 t > 0, (15) u(l, t) = X(L)T (t) = 0 = X(L) = 0 t > 0.(16)
17 There are three possible cases for the constant k: k = b 2 > 0 (trivial solution) X(x) = c 1 cosh bx + c 2 sinh bx, X(0) = X(L) = 0 = c 1 = c 2 = 0. k = 0 (trivial solution) X(x) = c 1 x + c 2, X(0) = X(L) = 0 = c 1 = c 2 = 0. k = β 2 < 0 (non-trivial solution) X(x) = c 1 cos βx + c 2 sin βx, X(0) = 0 = c 1 = 0 X(L) = 0 = β = β n = nπ L, n = 1, 2,...
18 A non-trivial solution for Eq. (13) is X(x) = X n (x) = c 2 sin nπ L x, n = 1, 2,..., with c 2 is arbitrary, and let say set c 2 = 1. By using k = β 2, solve Eq. (14) and get T (t) = T n (t) = b n cos λ n t + b n sin λ n t, n = 1, 2,..., Here λ n = cnπ L. Combining X n (x) and T n (t), we get a solution for Eq. (10) and also satisfies the Boundary Conditions Eqs. (11) u n (x, t) = X n (x)t n (t) = sin nπ L (b n cos λ n t+b n sin λ n t), n = 1, 2,... (17)
19 Principle of Superposition Theorem If φ and ψ are solutions to a linear differential equation and satisfy a linear boundary condition, then the linear combination u = c 1 φ + c 2 ψ is also a solution and satisfies the same boundary condition. Here c 1 and c 2 are constants. Since u 1 (x, t), u 2 (x, t),... are satisfying the 1D wave equation and the fixed ends boundary conditions. Thus, a general solution is the superposition of all these u n (x, t): u(x, t) = sin nπ L (b n cos λ n t + b n sin λ n t). (18) n=1
20 Half-range Fourier Series Applying initial condition u(x, 0) = b n sin nπ x = f(x), 0 < x < L, L k=1 L = b n = 2 f(x) sin nπ x dx, n = 1, 2,... L 0 L Applying initial condition u t (x, 0) = λ n b n sin nπ x = g(x), 0 < x < L, L = λ n b n = 2 L k=1 L 0 g(x) sin nπ x dx, n = 1, 2,... L
21 Summary Boundary-value problems 2 u t 2 = c 2 2 u, 0 < x < L, t > 0; t2 u(0, t) = 0 and u(l, t) = 0, t > 0; u(x, 0) = f(x) and u (x, 0) = g(x), 0 < x < L. t General solution u(x, t) = sin nπ L (b n cos λ n t + b n sin λ n t), b n = 2 L n=1 L 0 f(x) sin nπx L dx, b n = 2 λ n L L 0 λ n = cnπ L ; g(x) sin nπx L dx.
22 Sumarry for Method of Separation of Variables 1. Decompose u into products of functions of one variable. 2. Decompose the PDE into a set of ODEs. 3. Identify boundary conditions and the corresponding Sturm-Liouville problems. 4. Solve the Sturm-Liouville problems and obtain the corresponding eigenvalues. 5. Apply principle of superposition to obtain the eigenfucntion expansion of a general solution. 6. Use the initial conditions to obtain generalized Fourier coefficients of the eigenfunction expansion.
23 Examples 1. Let say a string with length L = 1 is fixed at two ends. The initial displacement of the string is f(x) = sin mπ L x and with zero initial velocity. Find u(x, t). 2. Same as the previous example, but { 3 10 f(x) = x, 0 x 1; 3 3(1 x) 1, x Now, assume that the initial displacement is 0, but the the initial velocity is g(x) = x cos x, given that L = 1 and c = 1. Find u(x, t).
24 D Alembert s Method (Optional) Boundary-value problems 2 u t 2 = c 2 2 u, 0 < x < L, t > 0; t2 u(0, t) = 0 and u(l, t) = 0, t; u(x, 0) = f(x) and u (x, 0) = g(x), 0 < x < L. t D Alembert solution u(x, t) = 1 2 [f (x ct)+f (x+ct)]+ 1 2c x+ct x ct where f and g are odd extension of f and g. g (s) ds, (19)
25 Examples (Optional) 1. Let say f(x) = sin mπ x and g(x) = 0, find the solution L for the 1D wave equation. 2. Same as the previous example, but L = 1, c = 1, π g(x) = 0, and { 3 10 f(x) = x, 0 x 1; 3 3(1 x) 1, x Now, let say L = 1, c = 1, f(x) = 0, g(x) = x, 0 < x < 1. Find the solution.
26 Two Dimensional Laplace Equation Problem: Consider a rectangle slab of length a and width b, assuming that it is properly insulated from top and bottom of the surfaces. The internal temperature distribution u(x, y, t), 0 < x < a, 0 < y < b, t > 0 in the slab in now given by 2D heat equation u = t c2 2 u. The four boundaries of the slab are kept at a constant temperature of zero for x = 0, x = a and y = 0, except at y = b where the temperature is kept at u(x, b, t) = f(x). Suppose the slab are left for a very long time, and the temperature distribution no longer changing with time, i.e. u = 0. t Now, find the steady state temperature distribution u(x, y) of the slab.
27 Boundary-value problems 2D Laplace Equation: 2 u x + 2 u = 0, 2 y2 0 < x < a, 0 < y < b; (20) Boundary Conditions: u(0, y) = u(a, y) = 0, 0 < y < a; (21) u(x, 0) = 0 and u(x, b) = f(x), 0 < x < a; (22)
28 Solving Laplace Equation 1. Decompose u into products of functions of one variable. u(x, y) = X(x)Y (y). 2. Decompose the PDE into a set of ODEs. X + kx = 0; (23) Y ky = 0. (24) 3. Identify boundary conditions and the corresponding Sturm-Liouville problems. X(0) = X(a) = 0, and Y (0) = 0. The Sturm-Liouville problem is X + kx = 0, X(0) = X(a) = 0.
29 4 Solve the Sturm-Liouville problems and obtain the corresponding eigenvalues. Non-trivial solution is Xn (x) = sin nπ a x, n = 1, 2,.... Corresponding solution for Y is Y n (y) = A n cosh nπ a y + B n sinh nπ a y. Apply BC Y (0) = 0, thus Yn (y) = B n sinh nπ a y. 5 Apply principle of superposition to obtain the eigenfucntion expansion of a general solution. u(x, y) = X n Y n = B n sinh nπ a y sin nπ a x. n=1 n=1 6 Use the initial conditions to obtain generalized Fourier coefficients of the eigenfunction expansion. nπb sinh a B n = 2 f(x) sin nπ x dx, n = 1, 2,.... a a a
30 Example Find the steady-state temperature distribution u(x, y) of a 1 2 slab, with y represent distance along the direction of the longer side of the slab. Given that one longer side of the slab is kept at 50 C and the other sides are kept at zero temperature.
31 Review on Boundary Conditions Review on Boundary Conditions
32 One Dimensional Heat Equation Again Recall the heat equation in our first example With the initial condition, u t = 2 u c2, 0 < x < L, t > 0. (25) x2 And the boundary condition, u(x, 0) = f(x), 0 < x < L. (26) u(0, t) = u(l, t) = 0, t > 0. (27) Here the boundary conditions are called Homogeneous Dirichlet s Boundary Conditions.
33 Homogeneous (Zero Temperature) Dirichlet s Boundary Condition We already showed that the solution for the zero temperature heat equation is u(x, t) = n=1 b n e λ2 n t sin nπ L x, n = 1, 2,..., (28) where b n = 2 L L 0 f(x) sin nπ L x dx. Now, let s take a closer look at the B.C.
34 Boundary conditions for 1D Heat Equation Three commonly used boundary conditions are Dirichlet s B.C. (the values of u are given on boundaries) (Homogeneous B.C.) u(0, t) = u(l, t) = 0, t > 0. (Non-homogeneous B.C.) u(0, t) = T 0, u(l, t) = T 1, t > 0 and T 0, T 1 0. Neumann s B.C. (normal derivatives are given on boundaries) u u x (0, t) = x (L, t) = 0, t > 0. Robin s B.C. (αu + u n are given on boundaries) This B.C. correspond to one end insulated and one end radiating heat. u(0, t) = 0, u x (L, t) = κu(l, t), t > 0. κ is called the convection coefficient.
35 Homogeneous (Non-Zero Temperature) Dirichlet s Boundary Condition u t = c 2 2 u, x2 0 < x < L, t > 0; I.C. : u(x, 0) = f(x), 0 < x < L; B.C. : u(0, t) = u(l, t) = T 0, t > 0. By changing the variable w(x, t) = u(x, t) T 0, we could recovered the zero temperature boundary-value problem w = c 2 2 w, 0 < x < L, t > 0; t x2 I.C. : w(x, 0) = f(x) T 0, 0 < x < L; B.C. : w(0, t) = w(l, t) = 0, t > 0.
36 Steady-State Solutions The steady-state solution, or time-independent solution, is when the change of temperature distribution u s (x, t) no longer depends on time t. This usually happens when t. In the steady-state situation, u = 0, and thus the heat t equation now becomes a second order ODE d2 u = 0. dx 2 If the boundary conditions are homogeneous u(0, t) = u(l, t) = T 0, then the steady-state solution is u s (x, t) = u s (x) = T 0. If the boundary conditions are non-homogeneous u(0, t) = T 0, u(l, t) = T 1, then the steady-state solution is u s (x, t) = u s (x) = T 1 T 0 x + T L 0.
37 Dirichlet s (Non-homogeneous) B.C. The corresponding boundary-value problem is u = c 2 2 u, 0 < x < L, t > 0; (29) t x2 B.C.: u(0, t) = T 0, u(l, t) = T 1, t > 0; (30) I.C.: u(x, 0) = f(x), 0 < x < L. (31) The strategy to solve PDE with non-homogeneous B.C. is Find the steady-state solution u s (x) that satisfies the B.C. Convert the non-homogeneous problem to a homogeneous problem by changing the variable w(x, t) = u(x, t) u s (x).
38 By substituting w(x, t) = u(x, t) u s (x) into the non-homogeneous B.C. problem, we get w = c 2 2 w, 0 < x < L, t > 0; t x2 I.C.: w(x, 0) = f(x) u s (x), 0 < x < L; B.C.: w(0, t) = w(l, t) = 0, t > 0. The solution to the homogeneous BVP is w(x, t) = b n e λ2nt sin β n x, where n=1 L b n = 2 (f(x) u s (x)) sin β n x dx. L 0 Finally, the solution to the non-homogeneous problem is u(x, t) = w(x, t) + u s (x).
39 Example Solve the following non-homogeneous boundary-value problem: u = 4 2 u, 0 < x < π, t > 0; t x2 I.C.: u(x, 0) = 50, 0 < x < π; B.C.: u(0, t) = 0, and u(π, t) = 100, t > 0.
40 Neumann s Boundary Conditions The corresponding boundary-value problem is u t = 2 u c2, 0 < x < L, t > 0; (32) x2 u u B.C.: (0, t) = (L, t) = 0, t > 0; (33) x x I.C.: u(x, 0) = f(x), 0 < x < L. (34) After separated the variables: X kx = 0, X (0) = X (L) = 0; T kc 2 T = 0.
41 Consider the three cases for the eigenvalue k: k = b 2 > 0 = X(x) = c 1 cosh bx + c 2 sinh bx, X (0) = X (L) = 0 = c 1 = c 2 = 0 trivial solution k = 0, = X(x) = c 1 x + c 2 X (0) = X (L) = 0 = c 1 = 0, c 2 arbitrary Let choose c2 = a 0 /2 where a 0 is a constant. k = β 2 < 0, = X(x) = c 1 cos βx + c 2 sin βx X (0) = 0 = c 2 = 0 X (L) = 0 = β = β n = nπ L, n = 1, 2,... βc1 is arbitrary, and set βc 1 to 1. The corresponding solution for T is T (t) = T n (t) = a n e λ2 n t.
42 Combining X n (x) and T n (t), and applying the principle of superposition, we get the general solution u(x, t) = a a n e λ2 n t cos nπ L n=1 x, n = 1, 2,..., (35) where a 0 = 2 L a n = 2 L L 0 L 0 f(x) dx, f(x) cos nπ L x dx.
43 Robin s Boundary Conditions The corresponding boundary-value problem is u t = 2 u c2, 0 < x < L, t > 0; (36) x2 u B.C.: u(0, t) = 0, (L, t) = κu(l, t), t > 0;(37) x I.C.: u(x, 0) = f(x), 0 < x < L. (38) After separated the variables: X kx = 0, T kc 2 T = 0. X(0) = 0, X (L) = κx(l);
44 Consider the three cases for the eigenvalue k: k = b 2 > 0 = X(x) = c 1 cosh bx + c 2 sinh bx, X(0) = 0, X (L) = κx(l) = c 1 = c 2 = 0 because κ, cosh bl and sinh bl are strictly positive. k = 0, = X(x) = c 1 x + c 2 X(0) = 0, X (L) = κx(l) = c 1 = c 2 = 0 trivial solution. k = β 2 < 0, = X(x) = c 1 cos βx + c 2 sin βx X(0) = 0 = c1 = 0 X (L) = κx(l) implies β must satisfies the non-linear equation β cos βl + κ sin βl = 0, which has infinite many roots, β = β n, n = 1, 2,.... Thus, X(x) = Xn (x) sin β n x.
45 The corresponding solution for T is T (t) = T n (t) = c n e λ2 nt, where λ n = cβ n. Combining X n (x) and T n (t), and applying the principle of superposition, we get the general solution u(x, t) = c n e λ2nt sin β n x, n = 1, 2,..., (39) n=1 where c n = 1 L 0 sin2 β n x dx L 0 f(x) sin β n x dx.
46 Example Solve the following Robin s problem: u = 2 u, 0 < x < 1, t > 0; t x2 I.C.: u(x, 0) = x(1 x), 0 < x < 1; B.C.: u u (0, t) = 0, and (1, t) = u(1, t), t > 0. x x
47 Two Dimensional Wave Equation Boundary-value problem ( ) 2 u 2 t = u 2 c2 x + 2 u, 0 < x < L, t > 0; 2 y 2 B.C.: u(0, y, t) = u(a, y, t) = 0, for 0 y b and t > 0; B.C.: u(x, 0, t) = u(x, b, t) = 0, for 0 x a and t > 0; I.C.: u(x, y, 0) = f(x, y) and u (x, y, 0) = g(x, y). t Separation of variables: u(x, y, t) = X(x)Y (y)t (t); u(0, y, t) = u(a, y, t) = 0 = X(0) = X(a) = 0; u(x, 0, t) = u(x, b, t) = 0 = Y (0) = Y (b) = 0.
48 T c 2 T = X X + Y Y = k2, gives T + c 2 k 2 T = 0; X = Y X Y k2 = µ 2 ; X + µ 2 X = 0 and Y + ν 2 Y = 0, where ν = k 2 µ 2. In summary X + µ 2 X = 0, X(0) = 0, X(a) = 0, Y + ν 2 Y = 0, Y (0) = 0, Y (b) = 0, Ẍ + c 2 k 2 T = 0, k 2 = µ 2 + ν 2.
49 Solution of the separated equations X(x) = c1 sin µx + c 2 cos µx; Y (y) = d1 sin νy + d 2 cos νy; T (t) = e1 sin ckt + e 2 cos ckt. where ν = k 2 µ 2. From the boundary condition for X and Y we get c2 = 0, µ = µ m = mπ a, and c 1 arbitrary (set to 1). d2 = 0, ν = ν n = nπ b, and d 1 arbitrary (set to 1). Thus, X(x) = Xm (x) = sin µ m x, m = 1, 2,... ; Y (y) = Yn (x) = sin ν n y, n = 1, 2,... ; T (t) = Tmn (t) = B mn cos λ mn t + B sin λ mn t, where λ mn = c µ 2 m + νn. 2
50 By using the principle of superposition, the solution to the 2D wave equation is u(x, y, t) = n=1 m=1 The coefficients are given by (B mn cos λt + Bmn sin λt) sin µ m x sin ν n y. b a Bmn = 4 f(x, y) sin µ m x sin ν n y dx dy; ab 0 0 B mn = 4 b a g(x, y) sin µ m x sin ν n y dx dy; abλ mn 0 0 µm = mπ a, ν n = nπ, m, n = 1, 2,.... b (40)
51 Two Dimensional Heat Equation Boundary-value problem u ( ) 2 t = u c2 x + 2 u, 0 < x < L, t > 0; 2 y 2 B.C.: u(0, y, t) = u(a, y, t) = 0, for 0 y b and t > 0; B.C.: u(x, 0, t) = u(x, b, t) = 0, for 0 x a and t > 0; I.C.: u(x, y, 0) = f(x, y), for 0 < x < a, 0 < y < b. General solution: u(x, y, t) = n=1 m=1 A mn sin µ m x sin ν n ye λmnt ; µ m = mπ, ν a n = nπ, λ b mn = c µ 2 m + νn; 2 A mn = 4 ab b a 0 0 f(x, y) sin µ m x sin ν n y dx dy.
Wave Equation With Homogeneous Boundary Conditions
Wave Equation With Homogeneous Boundary Conditions MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 018 Objectives In this lesson we will learn: how to solve the
More informationPartial Differential Equations
Partial Differential Equations Xu Chen Assistant Professor United Technologies Engineering Build, Rm. 382 Department of Mechanical Engineering University of Connecticut xchen@engr.uconn.edu Contents 1
More informationMethod of Separation of Variables
MODUE 5: HEAT EQUATION 11 ecture 3 Method of Separation of Variables Separation of variables is one of the oldest technique for solving initial-boundary value problems (IBVP) and applies to problems, where
More informationMath Assignment 14
Math 2280 - Assignment 14 Dylan Zwick Spring 2014 Section 9.5-1, 3, 5, 7, 9 Section 9.6-1, 3, 5, 7, 14 Section 9.7-1, 2, 3, 4 1 Section 9.5 - Heat Conduction and Separation of Variables 9.5.1 - Solve the
More informationProblem set 3: Solutions Math 207B, Winter Suppose that u(x) is a non-zero solution of the eigenvalue problem. (u ) 2 dx, u 2 dx.
Problem set 3: Solutions Math 27B, Winter 216 1. Suppose that u(x) is a non-zero solution of the eigenvalue problem u = λu < x < 1, u() =, u(1) =. Show that λ = (u ) 2 dx u2 dx. Deduce that every eigenvalue
More informationThe two-dimensional heat equation
The two-dimensional heat equation Ryan C. Trinity University Partial Differential Equations March 5, 015 Physical motivation Goal: Model heat flow in a two-dimensional object (thin plate. Set up: Represent
More informationSection 12.6: Non-homogeneous Problems
Section 12.6: Non-homogeneous Problems 1 Introduction Up to this point all the problems we have considered are we what we call homogeneous problems. This means that for an interval < x < l the problems
More informationName: Math Homework Set # 5. March 12, 2010
Name: Math 4567. Homework Set # 5 March 12, 2010 Chapter 3 (page 79, problems 1,2), (page 82, problems 1,2), (page 86, problems 2,3), Chapter 4 (page 93, problems 2,3), (page 98, problems 1,2), (page 102,
More information6 Non-homogeneous Heat Problems
6 Non-homogeneous Heat Problems Up to this point all the problems we have considered for the heat or wave equation we what we call homogeneous problems. This means that for an interval < x < l the problems
More informationSolutions to Exercises 8.1
Section 8. Partial Differential Equations in Physics and Engineering 67 Solutions to Exercises 8.. u xx +u xy u is a second order, linear, and homogeneous partial differential equation. u x (,y) is linear
More information1. Partial differential equations. Chapter 12: Partial Differential Equations. Examples. 2. The one-dimensional wave equation
1. Partial differential equations Definitions Examples A partial differential equation PDE is an equation giving a relation between a function of two or more variables u and its partial derivatives. The
More informationMATH 251 Final Examination August 14, 2015 FORM A. Name: Student Number: Section:
MATH 251 Final Examination August 14, 2015 FORM A Name: Student Number: Section: This exam has 11 questions for a total of 150 points. Show all your work! In order to obtain full credit for partial credit
More informationSolving the Heat Equation (Sect. 10.5).
Solving the Heat Equation Sect. 1.5. Review: The Stationary Heat Equation. The Heat Equation. The Initial-Boundary Value Problem. The separation of variables method. An example of separation of variables.
More informationThe lecture of 1/23/2013: WHY STURM-LIOUVILLE?
The lecture of 1/23/2013: WHY STURM-LIOUVILLE? 1 Separation of variables There are several equations of importance in mathematical physics that have lots of simple solutions. To be a bit more specific,
More informationPartial Differential Equations
Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives with respect to those variables. Most (but
More informationMATH-UA 263 Partial Differential Equations Recitation Summary
MATH-UA 263 Partial Differential Equations Recitation Summary Yuanxun (Bill) Bao Office Hour: Wednesday 2-4pm, WWH 1003 Email: yxb201@nyu.edu 1 February 2, 2018 Topics: verifying solution to a PDE, dispersion
More informationTHE METHOD OF SEPARATION OF VARIABLES
THE METHOD OF SEPARATION OF VARIABES To solve the BVPs that we have encountered so far, we will use separation of variables on the homogeneous part of the BVP. This separation of variables leads to problems
More informationv(x, 0) = g(x) where g(x) = f(x) U(x). The solution is where b n = 2 g(x) sin(nπx) dx. (c) As t, we have v(x, t) 0 and u(x, t) U(x).
Problem set 4: Solutions Math 27B, Winter216 1. The following nonhomogeneous IBVP describes heat flow in a rod whose ends are held at temperatures u, u 1 : u t = u xx < x < 1, t > u(, t) = u, u(1, t) =
More informationHeat Equation, Wave Equation, Properties, External Forcing
MATH348-Advanced Engineering Mathematics Homework Solutions: PDE Part II Heat Equation, Wave Equation, Properties, External Forcing Text: Chapter 1.3-1.5 ecture Notes : 14 and 15 ecture Slides: 6 Quote
More informationMATH 251 Final Examination December 19, 2012 FORM A. Name: Student Number: Section:
MATH 251 Final Examination December 19, 2012 FORM A Name: Student Number: Section: This exam has 17 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all
More informationMA 201: Method of Separation of Variables Finite Vibrating String Problem Lecture - 11 MA201(2016): PDE
MA 201: Method of Separation of Variables Finite Vibrating String Problem ecture - 11 IBVP for Vibrating string with no external forces We consider the problem in a computational domain (x,t) [0,] [0,
More informationMA 201, Mathematics III, July-November 2016, Partial Differential Equations: 1D wave equation (contd.) and 1D heat conduction equation
MA 201, Mathematics III, July-November 2016, Partial Differential Equations: 1D wave equation (contd.) and 1D heat conduction equation Lecture 12 Lecture 12 MA 201, PDE (2016) 1 / 24 Formal Solution of
More informationPartial Differential Equations Summary
Partial Differential Equations Summary 1. The heat equation Many physical processes are governed by partial differential equations. temperature of a rod. In this chapter, we will examine exactly that.
More informationLECTURE 19: SEPARATION OF VARIABLES, HEAT CONDUCTION IN A ROD
ECTURE 19: SEPARATION OF VARIABES, HEAT CONDUCTION IN A ROD The idea of separation of variables is simple: in order to solve a partial differential equation in u(x, t), we ask, is it possible to find a
More informationMATH 251 Final Examination May 4, 2015 FORM A. Name: Student Number: Section:
MATH 251 Final Examination May 4, 2015 FORM A Name: Student Number: Section: This exam has 16 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work must
More informationLecture 24. Scott Pauls 5/21/07
Lecture 24 Department of Mathematics Dartmouth College 5/21/07 Material from last class The heat equation α 2 u xx = u t with conditions u(x, 0) = f (x), u(0, t) = u(l, t) = 0. 1. Separate variables to
More informationPartial Differential Equations for Engineering Math 312, Fall 2012
Partial Differential Equations for Engineering Math 312, Fall 2012 Jens Lorenz July 17, 2012 Contents Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131 1 Second Order ODEs with Constant
More informationMath 201 Assignment #11
Math 21 Assignment #11 Problem 1 (1.5 2) Find a formal solution to the given initial-boundary value problem. = 2 u x, < x < π, t > 2 u(, t) = u(π, t) =, t > u(x, ) = x 2, < x < π Problem 2 (1.5 5) Find
More informationSC/MATH Partial Differential Equations Fall Assignment 3 Solutions
November 16, 211 SC/MATH 3271 3. Partial Differential Equations Fall 211 Assignment 3 Solutions 1. 2.4.6 (a) on page 7 in the text To determine the equilibrium (also called steady-state) heat distribution
More informationMidterm Solution
18303 Midterm Solution Problem 1: SLP with mixed boundary conditions Consider the following regular) Sturm-Liouville eigenvalue problem consisting in finding scalars λ and functions v : [0, b] R b > 0),
More informationBoundary Value Problems in Cylindrical Coordinates
Boundary Value Problems in Cylindrical Coordinates 29 Outline Differential Operators in Various Coordinate Systems Laplace Equation in Cylindrical Coordinates Systems Bessel Functions Wave Equation the
More informationMA Chapter 10 practice
MA 33 Chapter 1 practice NAME INSTRUCTOR 1. Instructor s names: Chen. Course number: MA33. 3. TEST/QUIZ NUMBER is: 1 if this sheet is yellow if this sheet is blue 3 if this sheet is white 4. Sign the scantron
More informationThe One-Dimensional Heat Equation
The One-Dimensional Heat Equation R. C. Trinity University Partial Differential Equations February 24, 2015 Introduction The heat equation Goal: Model heat (thermal energy) flow in a one-dimensional object
More informationDiffusion on the half-line. The Dirichlet problem
Diffusion on the half-line The Dirichlet problem Consider the initial boundary value problem (IBVP) on the half line (, ): v t kv xx = v(x, ) = φ(x) v(, t) =. The solution will be obtained by the reflection
More informationSeparation of variables in two dimensions. Overview of method: Consider linear, homogeneous equation for u(v 1, v 2 )
Separation of variables in two dimensions Overview of method: Consider linear, homogeneous equation for u(v 1, v 2 ) Separation of variables in two dimensions Overview of method: Consider linear, homogeneous
More informationENGI 9420 Lecture Notes 8 - PDEs Page 8.01
ENGI 940 Lecture Notes 8 - PDEs Page 8.01 8. Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives
More informationMath 2930 Worksheet Wave Equation
Math 930 Worksheet Wave Equation Week 13 November 16th, 017 Question 1. Consider the wave equation a u xx = u tt in an infinite one-dimensional medium subject to the initial conditions u(x, 0) = 0 u t
More informationSeparation of Variables. A. Three Famous PDE s
Separation of Variables c 14, Philip D. Loewen A. Three Famous PDE s 1. Wave Equation. Displacement u depends on position and time: u = u(x, t. Concavity drives acceleration: u tt = c u xx.. Heat Equation.
More informationMATH 251 Final Examination December 16, 2015 FORM A. Name: Student Number: Section:
MATH 5 Final Examination December 6, 5 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 5 points. In order to obtain full credit for partial credit problems, all work must
More informationReview For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: = 0 : homogeneous equation.
Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: y y x y2 = 0 : homogeneous equation. x2 v = y dy, y = vx, and x v + x dv dx = v + v2. dx =
More informationENGI 9420 Lecture Notes 8 - PDEs Page 8.01
ENGI 940 ecture Notes 8 - PDEs Page 8.0 8. Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives
More informationMath 311, Partial Differential Equations, Winter 2015, Midterm
Score: Name: Math 3, Partial Differential Equations, Winter 205, Midterm Instructions. Write all solutions in the space provided, and use the back pages if you have to. 2. The test is out of 60. There
More informationUndamped Vibration of a Beam
Undamped Vibration of a Beam Louie L. Yaw Walla Walla University Engineering Department PDE Class Presentation June 5, 2009 Problem - Undamped Transverse Beam Vibration +u p(x, t) 0 m(x), EI(x) +x V L
More informationFOURIER SERIES PART III: APPLICATIONS
FOURIER SERIES PART III: APPLICATIONS We extend the construction of Fourier series to functions with arbitrary eriods, then we associate to functions defined on an interval [, L] Fourier sine and Fourier
More informationFinal: Solutions Math 118A, Fall 2013
Final: Solutions Math 118A, Fall 2013 1. [20 pts] For each of the following PDEs for u(x, y), give their order and say if they are nonlinear or linear. If they are linear, say if they are homogeneous or
More information(The) Three Linear Partial Differential Equations
(The) Three Linear Partial Differential Equations 1 Introduction A partial differential equation (PDE) is an equation of a function of 2 or more variables, involving 2 or more partial derivatives in different
More informationM.Sc. in Meteorology. Numerical Weather Prediction
M.Sc. in Meteorology UCD Numerical Weather Prediction Prof Peter Lynch Meteorology & Climate Centre School of Mathematical Sciences University College Dublin Second Semester, 2005 2006. In this section
More informationPDE and Boundary-Value Problems Winter Term 2014/2015
PDE and Boundary-Value Problems Winter Term 2014/2015 Lecture 6 Saarland University 17. November 2014 c Daria Apushkinskaya (UdS) PDE and BVP lecture 6 17. November 2014 1 / 40 Purpose of Lesson To show
More informationHomework for Math , Fall 2016
Homework for Math 5440 1, Fall 2016 A. Treibergs, Instructor November 22, 2016 Our text is by Walter A. Strauss, Introduction to Partial Differential Equations 2nd ed., Wiley, 2007. Please read the relevant
More informationBranch: Name of the Student: Unit I (Fourier Series) Fourier Series in the interval (0,2 l) Engineering Mathematics Material SUBJECT NAME
13 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE UPDATED ON : Transforms and Partial Differential Equation : MA11 : University Questions :SKMA13 : May June 13 Name of the Student: Branch: Unit
More informationDifferential Equations
Differential Equations Problem Sheet 1 3 rd November 2011 First-Order Ordinary Differential Equations 1. Find the general solutions of the following separable differential equations. Which equations are
More informationStrauss PDEs 2e: Section Exercise 4 Page 1 of 6
Strauss PDEs 2e: Section 5.3 - Exercise 4 Page of 6 Exercise 4 Consider the problem u t = ku xx for < x < l, with the boundary conditions u(, t) = U, u x (l, t) =, and the initial condition u(x, ) =, where
More informationPartial Differential Equations
Partial Differential Equations Spring Exam 3 Review Solutions Exercise. We utilize the general solution to the Dirichlet problem in rectangle given in the textbook on page 68. In the notation used there
More informationLucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche
Scuola di Dottorato THE WAVE EQUATION Lucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche Lucio Demeio - DIISM wave equation 1 / 44 1 The Vibrating String Equation 2 Second
More informationProblem (p.613) Determine all solutions, if any, to the boundary value problem. y + 9y = 0; 0 < x < π, y(0) = 0, y (π) = 6,
Problem 10.2.4 (p.613) Determine all solutions, if any, to the boundary value problem y + 9y = 0; 0 < x < π, y(0) = 0, y (π) = 6, by first finding a general solution to the differential equation. Solution.
More informationMath 251 December 14, 2005 Answer Key to Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt
Name Section Math 51 December 14, 5 Answer Key to Final Exam There are 1 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning
More informationPartial Differential Equations Separation of Variables. 1 Partial Differential Equations and Operators
PDE-SEP-HEAT-1 Partial Differential Equations Separation of Variables 1 Partial Differential Equations and Operators et C = C(R 2 ) be the collection of infinitely differentiable functions from the plane
More informationBoundary Value Problems (Sect. 10.1). Two-point Boundary Value Problem.
Boundary Value Problems (Sect. 10.1). Two-point BVP. from physics. Comparison: IVP vs BVP. Existence, uniqueness of solutions to BVP. Particular case of BVP: Eigenvalue-eigenfunction problem. Two-point
More informationMATH 251 Final Examination December 16, 2014 FORM A. Name: Student Number: Section:
MATH 2 Final Examination December 6, 204 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 0 points. In order to obtain full credit for partial credit problems, all work must
More informationMcGill University April 20, Advanced Calculus for Engineers
McGill University April 0, 016 Faculty of Science Final examination Advanced Calculus for Engineers Math 64 April 0, 016 Time: PM-5PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer Student
More informationLecture 19: Heat conduction with distributed sources/sinks
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without explicit written permission from the copyright owner. 1 ecture 19: Heat conduction
More informationMathematical Modeling using Partial Differential Equations (PDE s)
Mathematical Modeling using Partial Differential Equations (PDE s) 145. Physical Models: heat conduction, vibration. 146. Mathematical Models: why build them. The solution to the mathematical model will
More information17 Source Problems for Heat and Wave IB- VPs
17 Source Problems for Heat and Wave IB- VPs We have mostly dealt with homogeneous equations, homogeneous b.c.s in this course so far. Recall that if we have non-homogeneous b.c.s, then we want to first
More informationVibrating Strings and Heat Flow
Vibrating Strings and Heat Flow Consider an infinite vibrating string Assume that the -ais is the equilibrium position of the string and that the tension in the string at rest in equilibrium is τ Let u(,
More informationIntroduction to Sturm-Liouville Theory and the Theory of Generalized Fourier Series
CHAPTER 5 Introduction to Sturm-Liouville Theory and the Theory of Generalized Fourier Series We start with some introductory examples. 5.. Cauchy s equation The homogeneous Euler-Cauchy equation (Leonhard
More informationCHAPTER 4. Introduction to the. Heat Conduction Model
A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL EQUATIONS
More informationLecture 21: The one dimensional Wave Equation: D Alembert s Solution
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without explicit written permission from the copyright owner. 1 Lecture 21: The one dimensional
More informationPlot of temperature u versus x and t for the heat conduction problem of. ln(80/π) = 820 sec. τ = 2500 π 2. + xu t. = 0 3. u xx. + u xt 4.
10.5 Separation of Variables; Heat Conduction in a Rod 579 u 20 15 10 5 10 50 20 100 30 150 40 200 50 300 x t FIGURE 10.5.5 Example 1. Plot of temperature u versus x and t for the heat conduction problem
More informationSAMPLE FINAL EXAM SOLUTIONS
LAST (family) NAME: FIRST (given) NAME: ID # : MATHEMATICS 3FF3 McMaster University Final Examination Day Class Duration of Examination: 3 hours Dr. J.-P. Gabardo THIS EXAMINATION PAPER INCLUDES 22 PAGES
More informationTHE UNIVERSITY OF WESTERN ONTARIO. Applied Mathematics 375a Instructor: Matt Davison. Final Examination December 14, :00 12:00 a.m.
THE UNIVERSITY OF WESTERN ONTARIO London Ontario Applied Mathematics 375a Instructor: Matt Davison Final Examination December 4, 22 9: 2: a.m. 3 HOURS Name: Stu. #: Notes: ) There are 8 question worth
More informationBoundary conditions. Diffusion 2: Boundary conditions, long time behavior
Boundary conditions In a domain Ω one has to add boundary conditions to the heat (or diffusion) equation: 1. u(x, t) = φ for x Ω. Temperature given at the boundary. Also density given at the boundary.
More informationWave Equation Modelling Solutions
Wave Equation Modelling Solutions SEECS-NUST December 19, 2017 Wave Phenomenon Waves propagate in a pond when we gently touch water in it. Wave Phenomenon Our ear drums are very sensitive to small vibrations
More informationChapter 10: Partial Differential Equations
1.1: Introduction Chapter 1: Partial Differential Equations Definition: A differential equations whose dependent variable varies with respect to more than one independent variable is called a partial differential
More informationMath 251 December 14, 2005 Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt
Math 251 December 14, 2005 Final Exam Name Section There are 10 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning of each question
More informationMATH 251 Final Examination May 3, 2017 FORM A. Name: Student Number: Section:
MATH 5 Final Examination May 3, 07 FORM A Name: Student Number: Section: This exam has 6 questions for a total of 50 points. In order to obtain full credit for partial credit problems, all work must be
More information3 Green s functions in 2 and 3D
William J. Parnell: MT34032. Section 3: Green s functions in 2 and 3 57 3 Green s functions in 2 and 3 Unlike the one dimensional case where Green s functions can be found explicitly for a number of different
More informationPhysics 6303 Lecture 8 September 25, 2017
Physics 6303 Lecture 8 September 25, 2017 LAST TIME: Finished tensors, vectors, and matrices At the beginning of the course, I wrote several partial differential equations (PDEs) that are used in many
More informationThe General Dirichlet Problem on a Rectangle
The General Dirichlet Problem on a Rectangle Ryan C. Trinity University Partial Differential Equations March 7, 0 Goal: Solve the general (inhomogeneous) Dirichlet problem u = 0, 0 < x < a, 0 < y < b,
More informationu tt = a 2 u xx u tt = a 2 (u xx + u yy )
10.7 The wave equation 10.7 The wave equation O. Costin: 10.7 1 This equation describes the propagation of waves through a medium: in one dimension, such as a vibrating string u tt = a 2 u xx 1 This equation
More informationBoundary value problems for partial differential equations
Boundary value problems for partial differential equations Henrik Schlichtkrull March 11, 213 1 Boundary value problem 2 1 Introduction This note contains a brief introduction to linear partial differential
More informationHomework 7 Solutions
Homework 7 Solutions # (Section.4: The following functions are defined on an interval of length. Sketch the even and odd etensions of each function over the interval [, ]. (a f( =, f ( Even etension of
More informationMath 220a - Fall 2002 Homework 6 Solutions
Math a - Fall Homework 6 Solutions. Use the method of reflection to solve the initial-boundary value problem on the interval < x < l, u tt c u xx = < x < l u(x, = < x < l u t (x, = x < x < l u(, t = =
More informationA Motivation for Fourier Analysis in Physics
A Motivation for Fourier Analysis in Physics PHYS 500 - Southern Illinois University November 8, 2016 PHYS 500 - Southern Illinois University A Motivation for Fourier Analysis in Physics November 8, 2016
More informationCHAPTER 10 NOTES DAVID SEAL
CHAPTER 1 NOTES DAVID SEA 1. Two Point Boundary Value Problems All of the problems listed in 14 2 ask you to find eigenfunctions for the problem (1 y + λy = with some prescribed data on the boundary. To
More informationLecture6. Partial Differential Equations
EP219 ecture notes - prepared by- Assoc. Prof. Dr. Eser OĞAR 2012-Spring ecture6. Partial Differential Equations 6.1 Review of Differential Equation We have studied the theoretical aspects of the solution
More information13 PDEs on spatially bounded domains: initial boundary value problems (IBVPs)
13 PDEs on spatially bounded domains: initial boundary value problems (IBVPs) A prototypical problem we will discuss in detail is the 1D diffusion equation u t = Du xx < x < l, t > finite-length rod u(x,
More informationINTRODUCTION TO PDEs
INTRODUCTION TO PDEs In this course we are interested in the numerical approximation of PDEs using finite difference methods (FDM). We will use some simple prototype boundary value problems (BVP) and initial
More informationc2 2 x2. (1) t = c2 2 u, (2) 2 = 2 x x 2, (3)
ecture 13 The wave equation - final comments Sections 4.2-4.6 of text by Haberman u(x,t), In the previous lecture, we studied the so-called wave equation in one-dimension, i.e., for a function It was derived
More informationFINAL EXAM, MATH 353 SUMMER I 2015
FINAL EXAM, MATH 353 SUMMER I 25 9:am-2:pm, Thursday, June 25 I have neither given nor received any unauthorized help on this exam and I have conducted myself within the guidelines of the Duke Community
More informationSturm-Liouville Theory
More on Ryan C. Trinity University Partial Differential Equations April 19, 2012 Recall: A Sturm-Liouville (S-L) problem consists of A Sturm-Liouville equation on an interval: (p(x)y ) + (q(x) + λr(x))y
More informationPhysics 6303 Lecture 9 September 17, ct' 2. ct' ct'
Physics 6303 Lecture 9 September 17, 018 LAST TIME: Finished tensors, vectors, 4-vectors, and 4-tensors One last point is worth mentioning although it is not commonly in use. It does, however, build on
More informationMATH 131P: PRACTICE FINAL SOLUTIONS DECEMBER 12, 2012
MATH 3P: PRACTICE FINAL SOLUTIONS DECEMBER, This is a closed ook, closed notes, no calculators/computers exam. There are 6 prolems. Write your solutions to Prolems -3 in lue ook #, and your solutions to
More informationSeparation of Variables
Separation of Variables A typical starting point to study differential equations is to guess solutions of a certain form. Since we will deal with linear PDEs, the superposition principle will allow us
More informationIntroduction of Partial Differential Equations and Boundary Value Problems
Introduction of Partial Differential Equations and Boundary Value Problems 2009 Outline Definition Classification Where PDEs come from? Well-posed problem, solutions Initial Conditions and Boundary Conditions
More information14 Separation of Variables Method
14 Separation of Variabes Method Consider, for exampe, the Dirichet probem u t = Du xx < x u(x, ) = f(x) < x < u(, t) = = u(, t) t > Let u(x, t) = T (t)φ(x); now substitute into the equation: dt
More informationAn Introduction to Partial Differential Equations
An Introduction to Partial Differential Equations Ryan C. Trinity University Partial Differential Equations Lecture 1 Ordinary differential equations (ODEs) These are equations of the form where: F(x,y,y,y,y,...)
More informationAnalysis III Solutions - Serie 12
.. Necessary condition Let us consider the following problem for < x, y < π, u =, for < x, y < π, u y (x, π) = x a, for < x < π, u y (x, ) = a x, for < x < π, u x (, y) = u x (π, y) =, for < y < π. Find
More informationA Guided Tour of the Wave Equation
A Guided Tour of the Wave Equation Background: In order to solve this problem we need to review some facts about ordinary differential equations: Some Common ODEs and their solutions: f (x) = 0 f(x) =
More informationSECTION (See Exercise 1 for verification when both boundary conditions are Robin.) The formal solution of problem 6.53 is
6.6 Properties of Parabolic Partial Differential Equations SECTION 6.6 265 We now return to a difficulty posed in Chapter 4. In what sense are the series obtained in Chapters 4 and 6 solutions of their
More informationMath 4263 Homework Set 1
Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that
More information