MATH 401 Study Guide - Midterm Guide: Dirichlet Problem, Fundamental Solution, Heat Equation

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9 Jan 2019

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Instructions: please read carefully: try as many questions as you can. It is possible to obtain a total of 125 points. A mark of n/125 will be treated as a mark of n/100: closed book exam. In particular, no notes or electronic devices are permitted. Find the eigenvalues to within 1% accuracy for the dirichlet problem y. Consider problem (i) given by u t u x u xx txf t. 0 x (a) set up the boundary value problem satisfied by the green"s function for problem. Consider problem (ii) given by q x u x u g x t. ,0 where in domain d, one has (ii). x. 0 (a) set up the boundary value problem satisfied by the green"s function for problem (b) express the solution. Consider simple closed curves of fixed length (cid:1) . Let a be the area enclosed by any such curve.

Related Questions

Help me with my diffeque homework

Suppose that the first and second derivatives of a function y=f(x) are given by: f'(x) = xe^x and f"(x) = e^x (x+1). Which of the following statements is true? a) f'(-1) > 0 b) f(x) is increasing and opens up (0, infinity) c) There are no critical points and there is one inflection point. d) f(x) is increasing and opens down on (negative infinity, -1) e) There are is one critical point and there are two inflection points. f) f"(x) > 0 on (negative infinity, -1) g) There are no critical points and there are two inflection points. h) f'(x) > 0 on (negative infinity, 0) i) f'(x) < 0 on (0, infinity) j) f"(x) < 0 on (-1, infinity)

Using a technique of integration and the separation-of-variables technique, you'll discover a function whose antiderivative models the shape of an ideal cable suspended from its endpoints. The function depends on a parameter b, which is determined by the properties of the cable, such as the density of the material out of which it is made.

Using principles from physics, it can be shown that when a perfectly flexible, uniformly dense, and inextensible cable is suspended from its endpoints, then it will assume a shape of a curve y = f(x) satisfying the following second-order differential equation:

where Ï is the density of the material of the cable, g is the acceleration due to gravity, and T is the magnitude of the tension in the cable at its lowest point.

We assume our cable is as in the diagram above and y = f(x) is its shape function; thus, not only does y satisfy the "hanging cable differential equation" above, we also have y '(0) = 0. Hence, making the substitution u =dy/dx in our the hanging cable equation, we obtain the first order, initial-value problem satisfied by u:

where b = Ïg/T Solve the preceding initial-value problem, entering your (explicit) solution in the form u = g(x) where g is a function of x in which the constant b is present.

MATH 401 Study Guide - Midterm Guide: Dirichlet Problem, Fundamental Solution, Heat Equation

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