(a) Find the function u which is harmonic outside the circle r - a and satisfies the Neumann cond...
4. Find the harmonic function in the exterior {r > a} of a sphere that satisfies the boundary condition - cos 0 on r a and which is bounded at infinity
4. Find the harmonic function in the exterior {r > a} of a sphere that satisfies the boundary condition - cos 0 on r a and which is bounded at infinity
7. (a) Find the harmonic function in the semi-infinite strip {0 < x < π, 0sy oo) that satisfies the "boundary conditions": u(r, y) (b) What would go awry if we omitted the condition at infinity?
7. (a) Find the harmonic function in the semi-infinite strip {0
A function u(x,y) is called harmonic if it satisfies Laplace's Equation: .Laplace's Equation is the driving force behind several types of physical models, including ideal fluid flow, electrostatic potentials, and steady-state distributions of heat in a conducting medium. Find TWO non-constant harmonic functions
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth harmonic For each x E D, the normal derivative of the function y K(x, y) . For each z e D, the function y K(x,y)-Г(z-y) is smooth near problem. This means the following: function on D(r satisfies (VyK(x, y).v(b))-arefor...
Find the area of the following region. The region outside the circle r = 2 and inside the circle r = - 4 cos 0 . The area of the region is square units. (Type an exact answer.)
4.9.74 For the following function f, find the antiderivative F that satisfies the given condition. f(u)=6e" - 7: F(0) = -1 The antiderivative that satisfies the given condition is F(u) =
The function u(x, t) satisfies the partial differential equation with the boundary conditions u(0,t) = 0 , u(1,t) = 0 and the initial condition u(x,0) = f(x) = 2x if 0<x<} 2(1 – x) if}<x< 1 . The initial velocity is zero. Answer the following questions. (1) Obtain two ODES (Ordinary Differential Equations) by the method of separation of variables and separating variable -k? (2) Find u(x, t) as an infinite series satisfying the boundary condition and the initial condition.
Find a vector function r that satisfies the following conditions. r"(t) = 7 cos 2ti + 5 sin 9tj + t?k, r(0) = i + k, r'(O) = i +j+k
c. Sketch the curve and find the area of the region that lies outside r 2sin0 and inside r=sin0+cos (you must use integral for this question, otherwise(like using formula for area of a circle, etc.,) you will get 0 point)
c. Sketch the curve and find the area of the region that lies outside r 2sin0 and inside r=sin0+cos (you must use integral for this question, otherwise(like using formula for area of a circle, etc.,) you will get 0 point)
Find the function y = y(2) (for x > 0) which satisfies the separable differential equation dy 6 + 14.2 dic 12 2 0 with the initial condition y(1) = 3. y=