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Unformatted text preview: 2.035: Selected Topics in Mathematics with Applications SOLUTIONS to the FINAL EXAM – Spring 2007 Problem 1: Using first principles (i.e. don’t use some formula like d/dx ( ∂F/∂φ ) − ∂F/∂φ = but rather go through the steps of calculating δF and simplifying it etc.) determine the function φ ∈ A that minimizes the functional 1 1 F { φ } = ( φ ) 2 + φφ + φ dx 2 over the set A = { φ  φ : [0 , 1] → R , φ ∈ C 2 [0 , 1] } . Note that the values of φ are not specified at either end. Solution: Taking the first variation of F gives 1 δF = φ δφ + δφφ + φδφ + δφ dx Integrating the terms involving δφ and δφ by parts yields 1 1 1 1 1 1 δF = φ δφ φ δφ dx + φ δφdx + φδφ φ δφ dx + δφ dx − − Two terms in this equation cancel out. We integrate the integral involving δφ by parts once more to get 1 1 1 1 1 δF = φ δφ + φδφ − φ δφ + φ δφ dx + δφ dx which simplifies to 1 1 δF = φδφ + φ + 1 δφ dx. Since this has to vanish for all arbitrary variations δφ ( x ) , < x < 1 , and for all arbitrary δφ (0) and δφ (1), it follows that the minimizer must satisfy the boundaryvalue problem φ + 1 = for ≤ x ≤ 1 , φ (0) = φ (1) = . The solution of this is readily found to be 1 φ ( x ) = x (1 − x ) for ≤ x ≤ 1 . 2 ——————————————————————————————————————– 1 Problem 2: A problem of some importance involves navigation through a network of sensors. Suppose that the sensors are located at fixed positions and that one wishes to navigate in such a way that the navigating observer has minimal exposure to the sensors. Consider the following simple case of such a problem. See figure on last page. A single sensor is located at the origin of the x, yplane and one wishes to navigate from the point A = ( a, 0) to the point B = ( b cos β, b sin β ). Let ( x ( t ) , y ( t )) denote the location of the observer at time t so that the travel path is described parametrically by x = x ( t ) , y = y ( t ) , ≤ t ≤ T . The exposure of the observer to the sensor is characterized by T E { x ( t ) , y ( t ) } = I ( x ( t ) , y ( t )) v ( t ) dt where v ( t ) is the speed of the observer and the “sensitivity” I is given by 1 I ( x, y ) = . x 2 + y 2 Determine the path from A to B that minimizes the exposure. Remark: For further background on this problem including generalization to n sensors, see the paper Qingfeng Huang, ”Solving an Open Sensor Exposure Problem using Variational Calculus”, Technical Report WUCS031, Washington University, Department of Computer Science and Engineering, St. Louis, Missouri, 2003 that I emailed you earlier in the term....
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 Spring '07
 RohanAbeyaratne
 Boundary value problem, φ, Boundary conditions, Euler equation

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