Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms Fu(x, t) _ u(x, t) + g(x) =-a ди (x, t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u(x, 0) 0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following...
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 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u (x,0)-0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following ODE where G() is the Fourier transform of g(x) and U(w,...
PDE. Please show all steps in detail.
2. Consider the 1D heat equation in a rod of length with diffusion constant Suppose the left endpoint is convecting (in obedience to Newton's Law of Cooling with proportionality constant K-1) with an outside medium which is 5000. while the right endpoint is insulated. The initial temperature distribution in the rod is given by f(a)- 2000 -0.65 300, 0<
please do not do question 1 but add "Assume...." conditions to #1,
thank you, upvote for sure
Consider the heat equation on "half-line" 0 <<< with prescribed zero temperature at x = 0 U = kuzx (0,t) = 0, ,0) = f(x) [the initial temperature). Find solution of this boundary value problem in the form u(x, t) = G(x – y, t) – 6(x + y,0)) f(y)dy, where G(x, t) = ome. Hint: Extend f(x) as an odd function f(x)...
2. In lecture, we talked about the heat equation on a thin, laterally insulated rod. There are many other domains on which you might want to determine the temperature. In this question, we explore the temperature on a wire that has been formed into a circle. thin wire, length 2L, laying flat on [-L,L] bend wire into a circular shape result is a circular wire where the ends x=L and x=-L correspond to one point now. While the PDE remains...
2. Consider the heat equation on a bounded domain with a zero heat-flux condition, 0<a <1 t > 0, u(z,0) = 2(1-2), (0, t) = 0, 14(1, t) = 0, t >0, t > 0, where σ > 0 is a constant. Such an equation is a model for the distribution of head throughout a rod which is thermally insulated on both ends. (a) Find the solution of the above PDE using separation of variables. You may use anything we...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0
1. Consider the Partial Differential Equation ot u(0,t) =...
(1 point) Consider the wave equation 1(1)utt = uzz for-oo < z < oo, t>0 with initial conditions ut (z,0-0 and u(z,0) = /(z), where (2) f(z) = 1 for 0 < z < 1, (3) f(z) =-1 for-1 < z < 0, and (4) f(z) = 0 for all other. The slanting lines in the figure below show the characteristics for this PDE that originate on the z-axis at the points of discontinuity of the initial data f f(x)...
1. Solve the Cauchy problem (2.1)-(2.2) for the following initial condition a) $(x) = 1 if |2<1 and $(x) = 0 if |z| > 1. b) p(x) = e-x, x > 0; $(x) = 0, x < 0. with the heat, or diffusion, equation on the real line. That is, we We begin with the hea sider the initial value problem Ut = kuxx, XER, t > 0, u(x,0) = 0(2), XER. (2.1) (2.2)