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),...
Please solve the above PDE by separation of variables and please show all steps y*u3 + 1?и; — 2 2 2 2 (гуu)? 2
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?
Solving PDE with separation of variables 3. Solve the heat flow equation on a circle. (10 point) Otu(t,0) = o u(t,0). such that the initial condition is u(0,0) = cos? (0)
Solve the heat equation by the method of separation of variables 1(1, t) = 0 Эт u,(0, t) = 0, u(x,0) =-2cos( 12. Solve the heat equation by the method of separation of variables 1(1, t) = 0 Эт u,(0, t) = 0, u(x,0) =-2cos( 12.
2. Consider the pde 0 <а < о, w(z,0) — 0, w(0, t) - t> 0, xwf = 0, = t Wr = (a) Use separation of variables to show that w(x, t) exp(k(t where k is a constant. (b) Show that the above solution does not satisfy both the initial and boundary conditions. (c) Use Laplace Transforms to solve the above pde. 2. Consider the pde 0
3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0 < x< 1 with boundary conditions ux(0, t) = 0 and ux(1, t) yields the general solution, 1, 0<x < 1/2 0, 1/2 x<1 Determine the coefficients An (n = 0, 1, 2, . . .) when u(x,0) = f(x) = 3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0
1 point) Solve the nonhomogeneous heat problem ut=uxx+4sin(2x), 0<x<π,ut=uxx+4sin(2x), 0<x<π, u(0,t)=0, u(π,t)=0u(0,t)=0, u(π,t)=0 u(x,0)=5sin(5x)u(x,0)=5sin(5x) u(x,t)=u(x,t)= Steady State Solution limt→∞u(x,t)=limt→∞u(x,t)= Please show all work. (1 point) Solve the nonhomogeneous heat problem Ut = Uxx + 4 sin(2x), 0< x < , u(0,1) = 0, tu(T, t) = 0 u(x,0) = 5 sin(52) u(a,t) Steady State Solution limt u(x, t) = Note: You can earn partial credit on this problem. Preview My Answers Submit Answers You have attempted this problem 0 times. You have unlimited attempts...
3. Using separation of variables to solve the heat equation, u -kuxx on the interval 0x<1 with boundary conditions u(0 and ur(1, t)-0, yields the general solution, u(x, t) =A0 + Σ Ane-k,t cos(nm) (with A, = ㎡π2) 0<x<l/2 0〈x〈1,2 u(x,0)=f(x)-.., , . . .) when u(x,0) = f(x)- Determine the coefficients An (n - 0, 1,2,
Please show all work and provide and an original solution. We can apply the Method of Separation of Variables to obtain a representation for the solution u u(, t) for the following partial differential equation (PDE) on a bounded domain with homogeneous boundary conditions. The PDE model is given by: u(r, 0) 0, (2,0) = 4. u(0,t)0, t 0 t 0 (a) (20 points) Assume that the solution to this PDE model has the form u(x,t) -X (r) T(t). State...
20. Given the associated PDE, demonstrate your understanding of separation of variables by creating two homogeneous ODEs. Do not solve. 36 = 977 +u, 0 < x < 1,0 <t< 4