u(x,0)=g(x)=0, du/dt(x,0)= h(x)= 6x(4-x).
Solution is u(x,t)= summation g(n,t) sinnπx/4,
g(n,t)= Cn sin(nπt)/4 + Dn cos(nπt)/4
Dn is 0 as g(x)=0 , h(x)= summation Cn Sin nπx/4
This implies Cn = 384{1-(-1)^n}/ π³n³
So, g(n,t) = (384{1-(-1)^n}) Sin(nπt/4)/ (π³n³ )
Consider the below wave equation with the given conditions. au 81 Ox? u(0,1) het au 0...
5. Given the initial-boundary value problem as below: ди ди at +u=k 0<x<1, 1>0, Ox?? Ou -(0,1) Ox Ou (1,t)=0, @x t>0, u(x,0) = x(1 - x) 0<x<1. where k is a non-zero positive constant. (i) By separation of variables, let the solution be in the form u(x,t) = X(x)T(t), show that one can obtain two differential equations for X(x) and T(t) as below: X"-cX = 0 and I' + (1 - ck)T = 0) where c is a constant....
Problem #9: Consider the below wave equation with the given conditions. clu olu 16 0<x< 5, t > 0, Ox2 u(0, 1) = u(5, 1) = 0, t > 0 700 u(x, 0) = 0, -0 = 7x(5– x) = Σ {1-(-1)"} sin(ntx/5), 0< x < 5. n=1 The solution to the above boundary-value problem is of the form U (x, t) = g(n, t) sin 97 * n=1 Find the function g(n, t). Problem #9: Enter your answer as...
3. [4] The solution of the wave equation02, which satisfies the boundary conditions u(0,t) = u(2,t) 0, is given by a cos+b sin If u(, t) satisfies the initial conditions u(x, 0)-0 and u(x,0)3sin(Tx) - sin(3T), find the coefficients an and bn Solution: b2 = , bs =- π 97T bn-0 otherwise, and an - 0 for all n 21. 3. [4] The solution of the wave equation02, which satisfies the boundary conditions u(0,t) = u(2,t) 0, is given by...
The conductive heat transfer in a rod of length L is described by the equation au ди əraat ,0<r<L,+20 where u(x, t) is the local temperature of the rod, t is time, and a is a positive constant describing the thermal conductivity of the rod. The initial and boundary conditions are: T(r, 0) = 0, T(L, t) = 0, and T (0, 1) = 1 for > 0 (1) Find the general solution of this PDE. (11) Find the eigenvalues...
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
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...
For 0 x π , 0S9, π , and 120 , solve the 2-D wave equation subject to the following conditions. u(0,y,t)-0, u(T.yt):0, u(x,0,) u(x,π, t) 0, 0 Boundary condition: C11 1 u(x),0)-sin(x)sin(2y) + sin(2x)sin(4y), 0 at It=0 Initial condition: For 0 x π , 0S9, π , and 120 , solve the 2-D wave equation subject to the following conditions. u(0,y,t)-0, u(T.yt):0, u(x,0,) u(x,π, t) 0, 0 Boundary condition: C11 1 u(x),0)-sin(x)sin(2y) + sin(2x)sin(4y), 0 at It=0 Initial condition:
Problem 2.7.26. Solve the parabolic problem ubject to the nonhomogeneous boundary conditions u(t,0)-1 and u(t,1)or 0 and the initial condition u(0,x)(x for xE(0,1) for some given function f:(0,1) R. Problem 2.7.26. Solve the parabolic problem ubject to the nonhomogeneous boundary conditions u(t,0)-1 and u(t,1)or 0 and the initial condition u(0,x)(x for xE(0,1) for some given function f:(0,1) R.
Consider the one-dimensional heat equation for nonconstant thermal properties debelo - (Kolon with the initial condition u(x, 0) = f(x). [Hint: Suppose it is known that if u(x, t) = ??(x ) h(t), then 1 dh 1 d do Ko(c) = -1 h dt c(x)p(x)o dx dx You may assume the eigenfuctions are known. Briefly discuss limt- u(x, t). Solve the initial value problem: (a) with boundary conditions u(0, t) = 0 and u(L, t) = 0 au *(b) with...
Let u be the solution to the initial boundary value problem for the Heat Equation au(t,) -48Fu(t,), te (0,oo), z (0,5); with boundary conditions u(t,0) 0, u(t,5) 0, and with initial condition 5 15 15 The solution u of the problem above, with the conventions given in class, has the form with the normalization conditions vn(0)-1, u Find the functions vnwn and the constants cn n(t) wnr) Let u be the solution to the initial boundary value problem for the...