4.2 Find the solution subject to the following boundary and initial conditions of the heat equation, ut = a²uxx- (d) u₂(0, 1) = -1 ift 20 u(1, t) + u₂(1,t) = 0 ift 20 u(a,0) = 0 if 0 ≤ x ≤ 1
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- ons als. 2. Solve the initial value problem x' = (t+1)/√t, x(1) = 4. 3. Find a function r(t) that satisfies the initial value problem r" = -3√t, x(1) = 1, r' (1) = 2.e 1 Use steps sizes (a) h = 0.1 and k = 0.0005 and (b) h = 0.1 and k = 0.01 to approximate the solution to the heat equation ди 22 и -(x, t) at √x²(x,t) = 0, 0 < x <1, 0≤t, with boundary conditions u(0,t) = u(1,t) = 0, 02. The following initial-boundary value problem for the heat equation. -2 cos p 0 0 u(0,t) = u(2,1) =1, t20 u(x,0) = sin 2ax + cos TX i. By using finite difference method with step size At = 0.01 and Ar = 0.4, show that Up1 =(-0.125 cos pu,1, +(1+0.25cos p)u,, - (0.125 cos plu1, i-1,t 11. Hence, by taking p = n, solve the heat equation above up to t= 0.01.Find the difference approximations of the solution y(x) of the boundary value problem y"+ 8(sin? πκ)y0, 0< x< 1 and y(0) = y(1) = 1 Use step-lengths h=1/4 and 1/6. Find an approximate value for y'(0). Show your outputs in terms of the following: 1. Paper computation and solutiondo Q1 2 ONLYPlease solve & show steps...4.2 Find the solution subject to the following boundary and initial conditions of the heat equation, ut = a²uxx- (b) u(0, t) = u(2,t) = 0, u(r,0) = 1 if 0Help me understand better1 ' The solution of the heat equation wzz =wt, 0Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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