Helpful Formulas: * Find the appropriate form of general solution for each of the following PDEs. (In cases, where 2. relevant find r) and w(r, 0) explicitly for the problem) (0,t)(7,t)0 Useful...
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) =...
PDE questions. Please show all
steps in detail.
2. Consider the initial-boundary value problem 0
Find the general solution by looking for solutions of the form , where r is a real or complex constant. Use the Equations of EULER-CAUCHY y(t) = + 12" + ty' +y=0,t> 0
Problem 6: I7 Points For the following periodic signal, x(t) 4OSesi a) Express the signal exponent +cos(9t) +2cos(15t) al in complex exponential Fourier series form. 13 r series coefficients and sketch the spectral line. [2 Find the fundamental frequency and identilY the harmonics in the signal. 12) Solution
Problem 6: I7 Points For the following periodic signal, x(t) 4OSesi a) Express the signal exponent +cos(9t) +2cos(15t) al in complex exponential Fourier series form. 13 r series coefficients and sketch the...
Explain how does w(x) been solve
Decomposing inhomogeneous PDEs to facilitate the use of separation of variables Inhomogeneities may arise in the initial (ICs) or boundary (BCs) conditions, or in the PDE itself. A simple example is the falling of an elastic wire under gravity: ə?u ,02u at2 = Car2g If the ICs are: u(x,0) = f(x) and (x,0) = 0, and the BCs are: u(0,t) = 0 and u(L,t) = h(t), then there are three inhomogeneities in this equation:...
use the hint please
2. Show that the Dirichlet problem for the disc t(z,y): +y S R2), where f(0) is the boundary function, has the solution 0o aj COS 1 sin j 3-1 where a, and b, are the Fourier coefficients of f. Show also that the Poisson integral formula for this more general disc setting is R22 (Hint: Do not solve this problem from first principles. Rather, do a change of variables to reduce this new problem to the...
Mark which statements below are true, using the following Consider the diffusion problem u(0,t)=0, u(L,t)=50 where FER is a constant, forcing term Any attempt to solve this using separation of variables fails. This is because the PDE is not homogeneous. A more fruitful approach arises from splitting the solution into the sum of two u(z,t) = X(z)T(t) + us(z), where the subscript designates the function as the steady limit and does not represent a derlvative. BEWARE: MARKING A STATEMENT TRUE...
z(t) y(t) 2 0 (22 points) Find the general solution to3"(1)-Ax(t), where A = | 1-2 | , and i(t)-
(1 point) In general for a non-homogeneous problem " ()y r)y-f(x) assume that yi, ye is a fundamental set of solutions for the homogeneous problem y"+p(r)y' +(xy-0. Then the formula for the particular solution using the method of variation of parameters is are where W(z) is the Wronskian given by the determinant where ufe) and u ,-1-nent), d dz. NOTE When evaluating these indefinite integrals we take the arbitrary constant of integration to be zero. So we have- Wed and...
problem 5
l lbout 0 for a general solution to the given differential equation u, y(0) = 0, V,(0) = 1 . Your answer should include a grneral formula for the ncients. (Find a recursive relation. If possible find Vi and 1,2). 3: Chebyshev's equation i(y + p'y-0, where p is a constant. Find two linearly Independert series solutions yi and ya. (Hint: find the series solution to the differential equation at z-0 to factor ao and ai as we...