Module Code: MATH2650 Extended Coursework Module Title: Calculus of Variations School of Mathematics Semester Two 201920 Academic integrity: • The work that you submit must be your own work and must represent your own under- standing. • You may refer to lecture notes, books, and reputable websites to help you complete the work. • While formal academic referencing will not be expected, you must indicate which sources you have used and where you have relied on them. • You must not ask others to help you to complete your assessments. Extended coursework information: • There are 2 pages (six questions) to this extended coursework. • Please submit your solutions to this extended coursework on GradeScope by 2pm on 14 May. Gradescope is accessible via “Submit My Work” in Minerva. • Answer all questions, giving full details and justification for your answers. • The number of marks for each question is indicated. 1. [20 marks] (a) Give an example of a functional I[y]. Throughout parts (b) and (c) below, illustrate the definitions and concepts both with a general functional of the form I[y] = ∫ x2 x1 F (x, y, y′) dx, and with your specific example. (b) Define the terms first variation, second variation and functional derivative, using the general functional and your specific example from (a) to illustrate the definitions. Be sure to include any extra conditions or concepts that you need, explaining why these are required. (c) Define the term extremal. Derive in general the necessary conditions for I[y] to be a minimum, justifying the equations and inequalities carefully. Are the conditions sufficient? Use your example from (a) to illustrate the steps needed to find the minimum value of I[y], if there is one. (d) Prove that the shortest curve between two points in the plane is a straight line. 2. [10 marks] A light ray passes from a point A = (0, 1) to a point B = (xB, yB) in a medium where the speed of light is proportional to y. What path does the light take? Page 1 of 2 Turn the page over Module Code: MATH2650 Extended Coursework 3. [10 marks] Find the extremals y(x) and z(x) of the functional I[y, z] = ∫ pi 0 ( 2y2 + 2z2 + 5yz′ − y′z + y′2 + z′2 ) dx subject to boundary conditions y(0) = 0, y(pi) = 0, z(0) = 2, and z(pi) = 0. 4. [10 marks] Find the extremal y(x) of the functional I[y] = ∫ 1 0 y′2 dx subject to boundary conditions y(0) = y(1) = 0, and subject to the constraint J [y] = ∫ 1 0 y dx = 1. Find the value of I corresponding to your extremal. By evaluating I for a suitable alternative function, argue that the extremal minimises I. 5. [10 marks] Use the function y = sinx to estimate the lowest eigenvalue λ0 of y′′ + λx2y = 0, with y(0) = y(pi) = 0. Does y = x(pi − x) give a better estimate of the eigenvalue? How would you improve on these estimates? 6. [20 marks] The functional I[y] is given by I[y] = ∫ pi 0 ( 1 2 y2 − y′2 + 1 2 y′′2 − 1 3 y3 + 1 6 y4 ) dx subject to boundary conditions y(0) = y(pi) = M and y′(0) = y′(pi) = 0, and subject to the constraint J [y] = ∫ pi 0 y dx = piM. Here, M is a real number. Show that y = constant is an extremal. For which values of M does this extremal give a minimum for I? Explain why it is difficult to find the general solution of the Euler–Lagrange equation in this case. Page 2 of 2 End.































































































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