ANSWER:
PDE(partial differential equation):
In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multi-variable functions and their partial derivatives.
1. Solve the vibrating string problem PDE BC BC IC IC utt T.T Uz(0,t) = 0 u(1,t0 u(x,0cos(3T) 14(2.0) = x
What type of PDE is this? Solve PDE using separation of variables (show all the work and logic) 05 x u(x,0) 4sin(37r), u,(x,0) 2sin(57) 0sx 1,t 2 0
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
Solve the circularly symmetric vibrating membrane PDE given as u_tt = ∇^2*u BC : u(1, θ, 0) = 0, 0 < t < ∞ ICs : u(r, θ, 0) = J_0*(2.4r) − 0.25*J_0*(14.93r), 0 ≤ r ≤ 1 u_t(r, θ, 0) = 0 Solve the circularly symmetric vibrating membrane PDE given as Utt = Dau BC : u(1,0,0) = 0, 0<t< oo ICs : u(r,0,0) = J.(2.4r) – 0.25J(14.93r), 0 <r <1 Ut(r,0,0) = 0
Determine the time-independent solution to the following BVIVP consisting of the PDE Ot for 0 <t and -1 < < 1, the BCs y(-1,t) = finite y(1,t)=0 and for -1 < 1 and the ICs y(z, 0) = 0 for 0 t Determine the time-independent solution to the following BVIVP consisting of the PDE Ot for 0
Exercise 9. Solve the BVP a(0, t) = 0 u(r, t)-Uz(n, t) = 0 u(z, 0) = sin z 0<x<π, t>0, t >0 t > 0 (z,0) = 0 2
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...
(1 point) Solve the wave equation with fixed endpoints and the given initial displacement and velocity. a2 ,0<x<L, t > 0 a(0. t) = 0, u(L, t) = 0, t > 0 Ou Ot ηπα t) + B,, sin (m Now we can solve the PDE using the series solution u(r,t)-> An C computed many times: An example: t) ) sin (-1 ). The coefficients .An and i, are Fourier coefficients we have , cos n-1 sin(n pix/ L) dr...
(3) Solve the following BVP for the Wave Equation using the Fourier Series solution formulac (3a2 u(r, t) 0 u(0, t)0 u(T, t) 0 u(r, 0) sin(x)2sin(4r) 3sin(8r) (r, 0) 10sin(2x)20sin (3r)- 30sin (5r) (r, t) E (0, ) x (0, 0o) t >0 t > 0 1 (3) Solve the following BVP for the Wave Equation using the Fourier Series solution formulac (3a2 u(r, t) 0 u(0, t)0 u(T, t) 0 u(r, 0) sin(x)2sin(4r) 3sin(8r) (r, 0) 10sin(2x)20sin (3r)-...
3. Determine the discretization of the heat equation driven at the boundary, PDE IC (x,0) BC t E R+ u(0, r) =g(x),ur(1,1) = 0, 3. Determine the discretization of the heat equation driven at the boundary, PDE IC (x,0) BC t E R+ u(0, r) =g(x),ur(1,1) = 0,