2. Classify each of the following as elliptic, hyperbolic or parabolic: 30'u 4 d’u du (a)...
and determine in which regions the given PDE’s are elliptic/ hyperbolic/ parabolic Problem 3 Find all solution of the following PDEs 1. 2. --= y, u=u(z, y). Problem 3 Find all solution of the following PDEs 1. 2. --= y, u=u(z, y).
Consider the following partial differential equation. au, au ax? + = u ay? Identify A, B, and C in the above equation and use them to calculate the following. B2 - 4AC = -1 + u X Classify the given partial differential equation as hyperbolic, parabolic, or elliptic. O hyperbolic parabolic elliptic
Consider the following partial differential equation. 22²ua²u at? ax² Identify A, B, and C in the above equation and use them to calculate the following. B2 - 4AC = x Classify the given partial differential equation as hyperbolic, parabolic, or elliptic. hyperbolic parabolic elliptic
3. A general surface of revolution is r(u, θ)-(f( u') cos θ , f(u) sin θ, υ), θΕ[0, 27), where f(u) is a positive function. For the following choices of f(u), find the principal, Gaus- sian, and mean curvatures at arbitrary (u, θ), and classify each point on the surface as elliptic, hyperbolic, parabolic, or planar. (a) f(u)u, u E [0, 00) (b) f(uV1 - u2,u e[-1,1].
b) i. Form partial differential equation from z = ax - 4y+b [4 marks] a +1 ii. Solve the partial differential equation 18xy2 + sin(2x - y) = 0 дх2ду c) i. Solve the Lagrange equation [4 Marks] az -zp + xzq = y2 where p az and q = ду [5 Marks] x ax ii. A special form of the second order partial differential equation of the function u of the two independent variables x and t is given...
11. Consider the parabolic coordinate system (u, v) related to the Cartesian coordi- nates (r, y) by х — 2иv, y — u? — u? for (и, v) € [0, оо) х [0, оо) 1 u = 1, u 2' (a) Sketch in the ry-plane the curves given u = 2. Then sketch in 1 v = 1, v = 2. Shade in the region R the xy-plane the curves given v = 2' bounded by the curves given by...
a. Name one physical law and one experimental law that are utilized in formulating the problem for determining the temperature distribution in a metal slab which is supplied heat at one end. b. State (i) the order of the following equation, (ii) whether it is linear or nonlinear, and (iii) if it is elliptic, .22u au 2 hyperbolic or parabolic: Əzz + axay + Lava 4+u = 0 c. Using the table provided in the formula sheet, or otherwise, find...
solve problem #1 depending on the given information Consider the following 1D second order elliptic equation with Dirichlet boundary conditions du(x) (c(x)du ) = f(x) (a $15 b), u(a) = ga, u(b) = gb dr: where u(x) is the unknown function, ga and gb are the Dirichlet boundary values, c(x) is a given coefficient function and f(x) is a given source function. See the theorem 10.1 in the textbook for the existence and uniqueness of the solution. 1.1 Weak Formulation...
DU .U . U U . 1). . . . 20. B={(-3, 2), (8, 4); and B' ={(-1.2), (2,-2); are two bases for R (a) Find the transition matrix from B' to B. (b) Find the transition matrix from B to B'. (c) let [V]8. = [-] find [V]
Consider the linear second-order PDE for u = u(x, y), 2uxx – 3uxy – 2uyy = (2x + y)2. (i) Determine the type (elliptic, hyperbolic, or parabolic) of (*). (ii) Introduce new independent variables s, t via x = 8 + 2t, y = -2s+t, and let w = w(s, t) be the function u in these new variables, i.e., let w(s, t) = u(s + 2t, -2s +t). Utilizing the chain rule, Ug = W58x + witz, Uy =...