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We can apply the Method of Separation of Variables to obtain a representation for the solution u ...
Apply the method of separation of variables to the PDE below to derive a pair of...
Apply the method of separation of variables to the PDE below to derive a pair of ODEs, one of which involves only x and the other of which involves only y. (You do not need to solve the ODE.) 23 u дх3 + x 23 u dy3 = 0 6 u=o L10)=0 Cha: Supplemental information -Linearity satisfies the property Leau, uz)=C.L(ui) +C₂L(42) - Heat Egn. is a linear partial differential equation : L(a)= eu-kay = f(xt) Linear homogeneous = L()...
Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0,...
Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0, (A) and the following boundary value/initial conditions: Ux(t,0) = uſt, 5) = 0, t>0, u(0, x) = 44(0, x) = 4 cos’ x, 0<x< (BC) (IC) for the function u= u(t, x). a. (5 points) Find ordinary differential equations for functions T = T(t) and X = X(x) such that the function u(t, x) = T(t)X(x) satisfies the PDE (A). b. (5 points) Find...
3. Consider the non homogeneous heat equation ut- urr+ 1 with non homogeneous boundary conditions u(0. t) 1, u(1t) (a) Find the equilibrium solution ueqx) to the non homogeneous equation. (b) The sol...
3. Consider the non homogeneous heat equation ut- urr+ 1 with non homogeneous boundary conditions u(0. t) 1, u(1t) (a) Find the equilibrium solution ueqx) to the non homogeneous equation. (b) The solution w(r, t) to the homogenized PDE wt-Wra, with w(0,t,t)0 1S -1 Verify that ugen(x, t)Ue(x) +w(x, t) solves the full PDE and BCs (c) Let u(x,0)- f(x) - 2 - ^2 be the initial condition. Find the particular solution by specifying all Fourier coefficients
3. Consider the...
9. Consider the beam PDE for the transverse deflection u(x, t) of an elastic beam Utt...
9. Consider the beam PDE for the transverse deflection u(x, t) of an elastic beam Utt + Kurz = 0 for 0 < x <L (30) where K > 0 is a constant. Suppose the boundary conditions are given by (31) u(0, t) = uz(0,t) = 0 Uwx (L, t) = Uzzz(L, t) = 0 (32) and the initial conditions are (33) u(x,0) = (x) u1(x,0) = V(x) (34) Use separation of variables to find the general solution to the...
solve the PDE +u= at2 on 3 € (0,L), t > 0, with boundary conditions au...
solve the PDE
+u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equation...
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0,...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0, 1] with boundary conditions u(0) = 0 and u(1) = 0, given by [-2 1 1-2 1 E RnXn h2 1 -2 1 This matrix can be considered a discrete version of the continuous operator d/da2 that acts upon a function(r). (a) Show that the n eigenvectors of A are given by the vectors ) (p-1,... , n) with components and with eigenvalues h2...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive in...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
2. Consider the initial-boundary value problem 0 0, t >0 Uz uェェ, Uz(0, t) _ u(0, t) = 0. a(z, 0...
PDE questions. Please show all
steps in detail.
2. Consider the initial-boundary value problem 0
Apply the method of separation of variables to the PDE below to derive a pair of ODEs, one of which involves only x and the other of which involves only y. (You do not need to solve the ODE.) 23 u дх3 + x 23 u dy3 = 0 6 u=o L10)=0 Cha: Supplemental information -Linearity satisfies the property Leau, uz)=C.L(ui) +C₂L(42) - Heat Egn. is a linear partial differential equation : L(a)= eu-kay = f(xt) Linear homogeneous = L()...
Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0, (A) and the following boundary value/initial conditions: Ux(t,0) = uſt, 5) = 0, t>0, u(0, x) = 44(0, x) = 4 cos’ x, 0<x< (BC) (IC) for the function u= u(t, x). a. (5 points) Find ordinary differential equations for functions T = T(t) and X = X(x) such that the function u(t, x) = T(t)X(x) satisfies the PDE (A). b. (5 points) Find...
3. Consider the non homogeneous heat equation ut- urr+ 1 with non homogeneous boundary conditions u(0. t) 1, u(1t) (a) Find the equilibrium solution ueqx) to the non homogeneous equation. (b) The solution w(r, t) to the homogenized PDE wt-Wra, with w(0,t,t)0 1S -1 Verify that ugen(x, t)Ue(x) +w(x, t) solves the full PDE and BCs (c) Let u(x,0)- f(x) - 2 - ^2 be the initial condition. Find the particular solution by specifying all Fourier coefficients
3. Consider the...
9. Consider the beam PDE for the transverse deflection u(x, t) of an elastic beam Utt + Kurz = 0 for 0 < x <L (30) where K > 0 is a constant. Suppose the boundary conditions are given by (31) u(0, t) = uz(0,t) = 0 Uwx (L, t) = Uzzz(L, t) = 0 (32) and the initial conditions are (33) u(x,0) = (x) u1(x,0) = V(x) (34) Use separation of variables to find the general solution to the...
solve the PDE
+u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0, 1] with boundary conditions u(0) = 0 and u(1) = 0, given by [-2 1 1-2 1 E RnXn h2 1 -2 1 This matrix can be considered a discrete version of the continuous operator d/da2 that acts upon a function(r). (a) Show that the n eigenvectors of A are given by the vectors ) (p-1,... , n) with components and with eigenvalues h2...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
PDE questions. Please show all
steps in detail.
2. Consider the initial-boundary value problem 0
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