Examination paper for TMA4122/TMA4123/TMA4125/TMA4130 Matematikk 4M/N

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1 Department of Mathematical Sciences Examination paper for TMA4122/TMA4123/TMA4125/TMA4130 Matematikk 4M/N Academic contact during examination: Markus Grasmair Phone: Code C): Basic calculator. Rottmann: Mathe- Examination date: August 2016 Examination time (from to): Permitted examination support material: matical formulæ Other information: All answers have to be justified, and they should include enough details in order to see how they have been obtained. All sub-problems carry the same weight for grading. Good luck! Language: English Number of pages: 3 Number pages enclosed: 2 Checked by: Date Signature

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3 TMA4122/TMA4123/TMA4125/TMA4130, Matematikk 4M/N Page 1 of 3 Problem 1 Only for TMA4125/TMA4130 Matematikk 4N! Use the Laplace transformation for solving the differential equation y + 3y + 2y = tu(t 1) with the initial conditions y(0) = 1, y (0) = 1. Problem 1 Only for TMA4122 Matematikk 4M! Let f be the function f(x) = x 2 for x R. Use the Fourier transformation for computing the convolution (f f)(x). Problem 1 Only for TMA4123 Matematikk 4M! Consider the Matlab script function x = TMA4123(N) x = 0 ; for i =1:N x = x (exp( x) x ^2)/(exp( x) 2 x ) ; end Compute the return value of the script for N = 2 and explain why this is an approximation to the solution of the equation e x = x 2.

4 Page 2 of 3 TMA4122/TMA4123/TMA4125/TMA4130, Matematikk 4M/N Problem 2 Find the polynomial of lowest degree that interpolates the points x i f(x i ) Problem 3 Use the trapezoidal rule with step length h = 0.25 in order to find an approximation T of the integral I = 1 Find an upper bound for the error I T. 0 e x2 dx. Problem 4 Perform two iterations of the Jacobi method for solving the linear system 5x 1 + 2x 2 + x 3 = 5, x 1 5x 2 + x 3 = 5, x 1 + x 2 + 3x 3 = 3. Use the initial value x (0) = (0, 0, 0). Problem 5 Let f be the 6-periodic function defined by Find the Fourier series of f. f(x) = x + 3 for 3 < x < 3. Problem 6 Let f be the 2π-periodic function given by e x for 0 < x < π, f(x) = 0 for π < x < 0. Assume that a n and b n are the Fourier coefficients of f, and denote by g and h the functions g(x) = a 0 + a n cos(nx) og h(x) = b n sin(nx). n=1 n=1 Sketch the graphs of the functions f, g, and h on the interval [ 2π, 2π], and find the values of f(x) and g(x) in the points x = π/2, x = 0, and x = π/2.

5 TMA4122/TMA4123/TMA4125/TMA4130, Matematikk 4M/N Page 3 of 3 Problem 7 equation We want to find a numerical solution of the partial differential t (x, t) = tu(x, t) + 2 u (x, t), x2 0 x 1, t > 0, with boundary conditions and initial condition u(0, t) = 0, u(1, t) = 1 for all t > 0 u(x, 0) = x for 0 x 1. Formulate an explicit method for solving this partial differential equation with the given boundary and initial conditions. Use a step length of h = 1/4 in space and perform two time steps of length k = 1/10. Problem 8 Use the Fourier transformation for solving the partial differential equation t (x, t) = u t 2 (x, t), x2 x R, t > 0, with initial condition u(x, 0) = e x2 2, x R. Problem 9 a) Given the equation 2 u x (x, y) + 2 u (x, y) + 5u(x, y) = 0, 0 < x < π, 0 < y < π/4, 2 y2 find all solutions of the form u(x, y) = F (x)g(y) that satisfy the boundary conditions u(0, y) = 0 and u(π, y) = 0, 0 < y < π/4. b) Find the solution of the problem in part a) that in addition satisfies the boundary conditions u(x, 0) = sin(x), 0 < x < π, u(x, π/4) = sin(x), 0 < x < π.

6 TMA4122/TMA4123/TMA4125/TMA4130, Matematikk 4M/N Page i of ii Fourier f(x) = 1 2π ˆf(ω)e iωx dω ˆf(ω) = 1 2π f(x)e iωx dx f g(x) 2π ˆf(ω)ĝ(ω) f (x) iω ˆf(ω) e ax2 1 2a e ω2 /4a e a x a a 2 +x 2 f(x) = 1 for x < a, 0 otherwise 2 a π ω 2 +a 2 π 2 e a ω 2 sin ωa π ω Laplace transform f(t) F (s) = 0 e st f(t) dt e at f(t) F (s a) cos(ωt) sin(ωt) cosh(ωt) sinh(ωt) s s 2 +ω 2 ω s 2 +ω 2 s s 2 ω 2 ω s 2 ω 2 t n n! s n+1 e at 1 s a f(t a)u(t a) δ(t a) e sa F (s) e as

7 Page ii of ii TMA4122/TMA4123/TMA4125/TMA4130, Matematikk 4M/N Numerics Newton s method: x k+1 = x k f(x k )/f (x k ) Newton s method for systems: JF(x (k) )h (k) = F(x (k) ) and x (k+1) = x (k) + h (k) with (JF(x (k) )) ij = j f i (x (k) ) Lagrange interpolation polynomial: L k (x) = (x x 1) (x x k 1 )(x x k+1 ) (x x n) (x k x 1 ) (x k x k 1 )(x k x k+1 ) (x k x n), p n (x) = n k=0 L k (x)f(x k ) Trapezoid rule: [ ] b a f(x) dx h 1 f f 1 + f f n 1 + 1f 2 n Error of the trapezoid rule: ɛ b a 12 h2 max a x b f (x). Simpson rule: b a f(x) dx [ ] h f0 + 4f f 2 + 4f f n 2 + 4f n 1 + f n with f i = f(x i ). Error of the Simpson rule: ɛ b a 180 h4 max a x b f (4) (x). Gauß Seidel iteration: x (k+1) = b Lx (k+1) Ux (k) with A = I + L + U. Jacobi iteration: x (k+1) = b + (I A)x (k) Euler method: y n+1 = y n + hf(x n, y n ) Improved Euler method: k 1 = hf(x n, y n ), k 2 = hf(x n + h, y n + k 1 ), y n+1 = y n k k 2. Classical Runge Kutte method: k 1 = hf(x n, y n ), k 2 = hf(x n + h/2, y n + k 1 /2), k 3 = hf(x n + h/2, y n + k 2 /2), k 4 = hf(x n + h, y n + k 3 ), y n+1 = y n k k k k 4. Backward Euler method: y n+1 = y n + hf(x n+1, y n+1 ) Finite differences: x (x, y) u(x+h,y) u(x h,y) 2h 2 u(x, y) u(x+h,y) 2u(x,y)+u(x h,y) x 2 h 2 u(x,y+h) u(x,y h) (x, y) y 2h 2 u(x, y) u(x,y+h) 2u(x,y)+u(x,y h) y 2 h 2