4. Show that the equation ut au bu du f(x, t) can be transformed into an...
please show all steps 4. Some equations that are not separable can be made separable by an appropriate substitution. Differential equations of the form y = f (!) are called Euler-homogeneous. These can be solved by letting v = y/t or y = ut. Using the product rule, dy du di = v + tai so that the differential equation becomes du v+t du f(0) - 0 dt t Use this technique to solve dy 2y+ + +4 - Leave...
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,...
au du atua = 90, дх with the initial conditions at t = 0: u=0 if u=-1-1 u=1 - << -1, if -1 <I<0, if 0 < I< 0. (Define u(r, t), x,t and the constant qo appropriately.) (b) Use the method of characteristics along suitable curves r(t) to obtain the implicit equation satisfied by the general solution ur,t) of the PDE given in the first problem (do not have to use the initial conditions at this stage, so there...
The equation y' 6x2 + 3y2 ту can be written in the form y' = f(y/x), i.e., it is homogeneous, so we can use the substitution u = y/x to obtain a separable equation with dependent variable u= u(x). Introducing this substitution and using the fact that y' = ru' + u we can write (*) as y' = xu'+u = f(u) where f(u) = Separating variables we can write the equation in the form dr g(u) du = where...
4) The one-dimensional heat equation is du 02u at = k→ ot ar 2 Where u(x,t)is a measure of the temperature at a location x on a bar at time t and k is a constant. Show by direct substitution thatu(x, t) = 10e-t sin x, satisfies the heat equation.
au + 6. The Korteweg-de Vries equation ди au + 6u 0 ( KV) at ax ar3 is an interesting model partial differential equation because two different physical effects are present: there is an expectation that solutions decay due to the third-order dispersive term; how- ever, the nonlinear term causes waves to steepen. Show that elementary traveling wave solutions of (KdV): u(x,t) = f(x - ct) yields an equation (5) corresponding to conservation of energy: 3(59)2 + fu? - bcf2...
P3. Solve the equation au(t, x) = kazu(t, x)-γυ(t, x) a(0, r) = f(x) f or-00 < x < oo with f E L(R), where k > 0 and γ E R. P3. Solve the equation au(t, x) = kazu(t, x)-γυ(t, x) a(0, r) = f(x) f or-00
please provide me with full working solution. Any help is appreciated. thank you in advance Consider the diffusion equation, au(x,t u(x,t) Here u(x,t) > 0 is the concentration of some diffusing substance, the spatial variable is x, time is t and D is a constant called the diffusivity with dimensions [LT-11. We will consider the diffusion equation on a finite spatial domain (0<x< 1) and an infinite time horizon (t > 0). To solve the diffusion equation we must include...
Need help with problem 4 au(x, t) cau(x,t) + 1Donu(x, t) Ot 3.5.4. Show that the diffusion equation (1.1.15) is of dissipative type.
9. Solve - cos(x) for 0 <x < 27, t > 0 ax2 at2 y(0, t) y(27, t) = 0 for t 0 y(x, 0) y(x.0)= 0 for 0 <x < 27. at Graph the fortieth partial sum for some values of the time. 11. Solve the telegraph equation au A Bu= c2- at ax2 at2 for 0 x < L, t > 0. A and B are positive constants The boundary conditions are u(0, t) u(L, t)=0 for t...