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4. (50 pts) Consider the following partial differential equation: 1du au Ət22 Ətər2 (BC) u7,t) =...
9) Solve the following partial differential equation au a2u ax2 n(0, t) = u(2, t)-0 t > 0 (x, 0) = 0 au at It=0 =x(2-x) 9) Solve the following partial differential equation au a2u ax2 n(0, t) = u(2, t)-0 t > 0 (x, 0) = 0 au at It=0 =x(2-x)
Consider the following partial differential equation. au, au ax? + = u ay? Identify A, B, and C in the above equation and use them to calculate the following. B2 - 4AC = -1 + u X Classify the given partial differential equation as hyperbolic, parabolic, or elliptic. O hyperbolic parabolic elliptic
Solve the following system of partial differential equations on - <r<0. u + 1x + 70, +6w 24-U: +3w, W -2 u(,0) v(3,0) w(1,0) = = = = = = 0. 0. 0. 10(E). (). (x).
Please help me with this ! Thank you 4. Consider the partial differential equation au au at + 0 Əx a) An explicit finite difference representation if this equation can be written as Ui,n+1 = Ui,n [Ui+1,1 – Ui-1,1] where 2 Perform a Von Neumann stability analysis to show that this finite difference equation is unconditioanlly unstable. Δt Δx b) By replacing the term Ui,n by the average of its neighboring values at the same time step, the following explicit...
2. Consider the following problem au au at2 = 2,2 -00<< ,> 0. 1- C for - 1<x<1 u(a,0) = 1 0 for x > 1 (3,0) = sin(x), -o0 < x < 00. Write the solution of the problem as a sum of a forward and backward wave.
1. Wave equation. Consider the wave equation on the finite interval (0, L) PDE BC where Neumann boundary conditions are specified Physically, with Neumann boundary conditions, u(r, t) could represent the height of a fluid that sloshes between two walls. (a) Find the general Fourier series solution by repeating the derivation from class now considering Neumann instead of Dirichlet boundary conditions. Your final solution should be (b) Consider the following general initial conditions u(x, 0)x) IC IC Derive formulas that...
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...
Need help with this problem. BC 1. Solve the vibrating string problem PDE Uz = 4uzz uz(0,t) = 0 ВС uz(1,t) = 0 IC u(a,0) = cos(372) (3,0) = r 0<x< 1, 0 <t< oo 0<t< 0<t<oo 0<x<1 0<x<1. IC
Consider the below wave equation with the given conditions. au 81 Ox? u(0,1) het au 0 < x < 4, t > 0, u(4,t) = 0, 1 > 0 op u(x,0) = 0, ди at = 6x(4- x) = 384 ${1 - (-1)"} sin(npox/4), 0< x < 4. n=1 The solution to the above boundary-value problem is of the form u(x,t) = 8(n, t) sin "* n=1 Find the function g(n,1).
2. Consider the following partial differential equation (a) Separate this equation into two ordinary differential equations (b) Translate the following boundary conditions on the above partial differential equation to conditions on the ordinary differential equations found above. 2. Consider the following partial differential equation (a) Separate this equation into two ordinary differential equations (b) Translate the following boundary conditions on the above partial differential equation to conditions on the ordinary differential equations found above.