MATH2018 Quiz The PDE ar2 can be solved using D'Alembert method. That is, it has a solution of the form u(x, t) = φ(x + ct) + ψ(x-ct). where c 6 Solve the PDE with the initial conditions u(x, 0)...
statisfied the PDE Suppose that λ-λ 1,22 are constants such that u(x, y)-F(x + λy) 24 u(x,y))-10 ya for any twice-differentiable function of one variable F(s) Suppose also that λ1 λ2 , enter the values of λ 1, λ2 in the boxes below. 6 2 Hence the soution will be of the form Solve the PDE with the initial conditions u (x, 0)-5 sin (a), lty (x,0)-0. Enter the expression for u(x, y) in the box below using Maple syntax...
2. In lectures we solved the heat PDE in 1 +1 dimensions with constant-temperature boundary conditions u(0,t)u(L,t) -0. If these boundary conditions change from zero temperature, we need to do a little bit more work. Consider the following initial/boundary-value problem (IBVP) 2 (PDE) (BCs) (IC) u(0,t) = a, u(x,00, u(L, t)=b, st. and let's take L = 1, a = 1, b = 2 throughout for simplicity. Solve this problem using the following tricks b and A"(x)-0 (a) Find a...
Problem # 3 [20 Points] Solve PDE: ut = uxx - u, 0 < x < 1, 0 < t < ∞ BCs: u(0, t)=0 u(1, t)=0 0 < t < ∞ IC: u(x, 0) = sin(πx), 0 ≤ x ≤ 1 directly by separation of variables without making any preliminary trans- formation. Does your solution agree with the solution you would obtain if transformation u(x, t)= e(caret)(-t) w(x, t) were made in advance?
9. Solve the wave equation subject to the boundary and initial conditions u(0,t) = 0, u(x,0) = 0, U(TT, t) = 0, t> 0 $ (3,0) = sin(x), 0<x<a
(2) Solve the following BVP for the Wave Equation using the Fourier Serics solution formulac (42- u(r, t)= 0 u(0, t) 0 u(5, t) 0 u(r, 0) sin(T) 12sin(T) ut(r, 0)0 (r, t) E (0,5) x (0, oo) t > 0 t > 0 (2) Solve the following BVP for the Wave Equation using the Fourier Serics solution formulac (42- u(r, t)= 0 u(0, t) 0 u(5, t) 0 u(r, 0) sin(T) 12sin(T) ut(r, 0)0 (r, t) E (0,5) x...
If you were to solve the variant of wave equation utt=uxx+u for 0<x<6 and t>0 with u(0,t)=u(2 ,t)=0, u(x,0)=2x, ut(x,0)=0 using separation of variables, what would be the correct form of Xn (x)? Xn (x)=cosh( nπ 4 Xn (x)=sin( nπ 2 Xn (x)=sin( n2 π2 4 Xn (x)=cos nπ 2 None of these
Solve the equation yu- xui = u, t > 0,x >0 with the initial conditions u(x, 0) =1 + x2 using the method of characteristics. Find the u(x, y). Substitute your found solution u(x, y) in the equation and verify that it satisfies the equation. solution explicitly in the form u =
(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...
please answer them in detail. The concentration C(x, t) of a pollutant in a semi-infinite pipe diffuses according to the PDE over 0<<o for > 0, where the diffusivity D is a given positive constant. An experiment is performed in which the initial concentration is zero everywhere, so that C(x, 0)0 for all r 20. Once t > 0 a fixed concentration of C(0, t) Co is imposed at x-0, where Co is a constant, and it is also assumed...
<---- equation (6) Using the equation (6) in page 569 of our text, obtain the solution of ut-CUx -lx in integral form satisfying the initial condition u(x, 0) = e- 30 1 | [A(p) cos px + B(p)sin px) e-c2p2tdp 11 (x, t; p) dp= u(x, t)= 0 Using the equation (6) in page 569 of our text, obtain the solution of ut-CUx -lx in integral form satisfying the initial condition u(x, 0) = e- 30 1 | [A(p) cos...