Variational and other methods

Transcription

1 Vritionl nd other methods Mrch 7, Find the function tht provides the extreml vlue for the following functionl I[y] = π/2 (y 2 y 2 + x 2 )dx, y() = y (π/2) = 1; y () = y(π/2) =. 2. Find the function tht provides the extreml vlue for the following functionl I[y(x)] = π/4 (y 2 y 2 +6y sin 2x)dx; y() =, y(π/4) = 1 3. Find the function tht provides the extreml vlue for the following functionl x 1 1+y 2 I[y(x)] = dx; y(x y 2 )=y,y(x 1 )=y 1 x 4. Find the function tht provides the extreml vlue for the following functionl I[y(x)] = (y 2 +2yy 16y 2 )dx; >, y() =, y() = 5. Find the function tht provides the extreml vlue for the following functionl I[y(x)] = (y 2 + y 2 +2y exp 2x)dx; y() = 1/3, y(1) = e 2 /3 1

2 6. Find the function tht provides the extreml vlue for the following functionl I[y(x)] = (y 2 2xy)dx; y() = y () =, y(1) = 1/12, y (1) is rbitrry 7. Find the function tht provides the extreml vlue for the following functionl I[y(x)] = 4 (y 1) 2 (y +1) 2 dx; y() =, y(4) = 2 8. Find the function tht provides the extreml vlue for the following functionl π/4 I[x] = (4x 2 x 2 +8x)dt x() = 1 x(π/4) = 9. Find the function tht provides the extreml vlue for the following functionl I[x] = (x 2 + x 2 +2x exp(2t))dt x() = 1/3 x(1) = e 2 /3 1. Find curve of given length for which the re of curviliner trpezoid ABCD is mximum, A =(x,y ), B =(x 1,y 1 ). 11. Show tht, if y stisfies Euler-Lgrnge equtions ssocited with the integrl x 2 I = (p 2 y 2 + q 2 y 2 )dx x 1 2

3 where p(x) nd q(x) re know function,i hs the vlue I = p 2 y y x 2 x Introduce convenient set of generlized coordintes nd derive the Lgrnge equtions of motion for single prticle of mss m moving in homogenous grvittionl field. The prticle is constrined to move on the surfce of given sphere of rdius R (sphericl pendulum). 13. Find the mximum volume of rectngulr prllelepiped tht is inscribed in n ellipsoid of semixes, b nd c. Find the rtio of the mximum volume of the prllelepiped to the volume of the ellipsoid. 14. In the method of the lest squres of regression theory the curve y = +bx is fit to the dt set (x i,y i ); i =1,..., N by requiring minim of the sum of squred errors N S(, b) = (y i ( + bx i )) 2 i=1 Find the expressions for nd b for given set (x i,y i ). 15. Find n pproximte solution of the problem of finding minimum of the functionl I[y] = (y 2 y 2 2xy)dx y() = y(1) = nd compre this solution with the exct solution Hint: An pproximte solution cn be determimed in the form y n = x(1 x)(α + α 1 x α n x n ) Perform clcultions for n =ndn = Derive the prtil differentil eqution for the rel field φ tht obeys the vritionl principle δ Ldxdt =, where the Lgrngin density is φ L = φ t x 2 µ2 φ λφ4. 3

4 17. Derive the prtil differentil eqution for the rel field φ tht obeys the vritionl principle δ Ld 3 xdt =, where the Lgrngin density is L = 1 2 φ2 φ 2 + m 2 φ Show tht tht the following Lgrngin density leds to the Mxwell equtions in vcuum L = 1 2 ε E 2 1 B 2. 2µ Hint: Use the vector nd sclr potentils to represent electric nd mgnetic field. 19. Show tht the following vritionl problem δ Ldt =, with the Lgrngin L(x, ẋ,t) = 1 2 mv2 qφ(x,t)+qv A(x.t), gives the equtions for prticle motion in the electromgnetic field. 2. Show tht tht the following Lgrngin density leds to the Mxwell equtions in mteril spce L = 1 2 ε E 2 1 B 2 ρφ. 2µ 21. Derive the Ohms lw J =σ(r)e(r) from the condition of the minimum of the totl energy Q dissipted in the conductive resistive medium J 2 (r) Q = σ(r) d3 r, V Hint: Consider the time independent cse so tht the chrge conservtion in the form J = cnbeusedsndditionlcontstrint. 22. Consider the expression J[u] = b b u(t)k(s, t)u(s)dtds + 4 b u 2 (s)ds 2 b f(s)u(s)ds,

5 where K(s, t) is symmetricl continuous function in the domin t b, s b, f(s) is given function of s, ndu(s) issoughtfunction. Find the Euler eqution for the function u(s) which delivers n extreml vlue to J[u]. 23. Electromgnetic oscilltions inside cvity resontor filled with ir re describedbythemxwellequtionintheform H ω 2 ε µ H = Formulte the vritionl form for the eigenvlue ω 2. Suggest simple one prmetric form of tril function nd evlute the lowest eigenvlue for the cylindricl cvity of rdius R. Compre it with the exct solution. Assume tht the mgnetic field is in the poloidl direction, H = H φ. 24. Show tht the Lgrngin (density) function L = y x y t 2y 3 x y 2 xx cn be used to derive the Korteveg-de-Vries eqution in the form where ϕ = y x. 25. Show tht the vritionl problem with the constrint I[φ] = G[φ] = ϕ t +3(ϕ 2 ) x + ϕ xxx = b b b is equivlent to the eigenvlue problem φ(x) =λ dxdtk(x, t)φ(x)φ(t) dxφ(x)φ(x)=constnt b dtk(x, t)φ(t). 26. Show tht the function h(u) tht minimizes the functionl W [h] W [h] = 2 h(u)f(u)du + stisfies the integrl eqution (Wiener-Hopf eqution) f(u) = h(u)g(u v)dv. h(u)h(v)g(u v)dudv Note tht g(u) is n even function. Hint: Consider the vrition of h(u) = h (u)+εq(u). 5

6 27. The stte of some system is described by continious vrible x(t), < t<t. This system is controlled by the externl control signl u(t) so tht the system is described by the eqution dx = x + u(t), dt where isthepositiveconstnt. Findthetrjectoryofthissystemfrom x = x() to x 1 = x(t ) if the control signl u(t) is chosen to minimize the integrl J = T (x(t) 2 + u(t)) 2 dt. Determine lowest eigenvlue λ to O(ε) for u xx +(1+εy)u yy + λu =, ε, <x<,<y<, u(x, ) = u(x, ) =u(, y)=u(, y) = 28. Find the extreml vlue of the functionl I[x] = [x sin(πx) (t + x) 2 ]dt 29. Using the Ryleigh-Ritz method find n pproximte solution of the following differentil eqution y + x 2 y = x, y() = y(1) =. You should construct the vritionl form first nd then minimize it with n pproprite tril function. Try to improve your solution by dding nother vritionl prmeter in your tril function. How does the vritionl form chnge if the boundry conditions re y() =, y(1) + y (1) =?. 3. Fundmentl modes of vibrtion of fixed wedge of constnt thickness cn be found by finding extrem of the functionl I[y] = (x 3 y 2 ωxy 2 )dx y(1) = y (1) = where ω is the vibrtion frequency, is given positive constnt. Use the tril function y =(x 1) 2 (α 1 + α 2 x)tofind two equtions tht define 6

7 the extrem of I[y]. Consider solubility condition for these equtions to determine eqution for ω in terms of. 31. Find n pproximte solution of the problem of extrem of the functionl I[y] = 2 1 (xy 2 x2 1 y 2 2x 2 y)dx x y(1) = y(2) = nd compre this solution with the exct solution. Hint: An pproximte solution cn be determined in the form 32. Consider the problem y = α(x 1)(x 2) 2 u + λu = εf(x, y, z)u, ε with u vnishing on the surfce of cube with side π. Determine the first order pproximtion for the lowest λ if f(,x,y,z)=x 2. Solve this problem lso by using the Ryleigh-Ritz technique with pproprite tril function. 33. Show tht the following eigenvlue problem u + λu =, <x<l u() =, u (l)+hu(l) =,h>, follows from the energy integrl subject of the constrint l I[u] = (u ) 2 dx + hu 2 (l) l u 2 dx = constnt. 7

8 34. Consider the following eigenvlue problem u + λu =, <x<1 u() =, u (1) + u(1) =, () Find the exct solution of this problem (b) Find the pproximte solution using the Ryleigh quotient with the tril function u 1 = x 2x 2 /3 (c) Find the pproximte solution using the Ryleigh-Ritz method with the tril function u 2 =sin(x) (d) Find the pproximte solution using the Ryleigh-Ritz method with the tril function u 3 = x + bx 2 (e) Compre ll three solutions 35. Obtin the pproprite form for the energy integrl for the following eigenvlue problem (pu ) + qu = λρu, <x<l u () h 1 u() =, u (l)+h 2 u(l) =, (h 1,h 2 ) >, 36. Use Ryleigh quotient to pproximte the leding eigenvlue for the following problem 2 u + λu =, x 2 + y 2 < 1 u =, x 2 + y 2 =1 Use the pproximte function φ 1 =cos(πr/2). 37. Use Ryleigh-Ritz method to solve the Dirichlet problem 2 u = u, x 2 + y 2 < 1 u = x 2, x 2 + y 2 =1 Use the pproximte functions φ = x 2 nd φ 1 =cos(πr/2). 38. Use Glerkin method to solve the Dirichlet problem 2 u u x + u y = 1, x 2 + y 2 < 1 u = x 2, x 2 + y 2 =1 Use the pproximte functions φ = x 2 nd φ 1 =cos(πr/2). 8

9 39. Solve the Dirichlet problem u xx + u yy = 1, x 2 + y 2 < 1 u = x 2, x 2 + y 2 =1, using the Ryleigh-Ritz method. Let u = u (x)+c 1 u 1 (x), where u = x 2 nd u 1 =cosπr/2. 4. Use vritionl method to obtin n upper bound on the lowest eigenfrequency of the following oscillting system 2 u x 2 x2 u = 2 u t 2, <x<1, u(,t)=u(1,t)=. 41. Find the pproximte solution to the following problem u xx + u yy = sinπx, <x<1, <y<1 u = on x =,x=1,y =,y =1 Use the following tril function where u = 1 φ 1 (x, y)+ 2 φ 2 (x, y)+ 3 φ 3 (x, y) φ 1 (x, y) = xy(1 x)(1 y) φ 2 (x, y) = x 2 y(1 x)(1 y) φ 3 (x, y) = xy 2 (1 x)(1 y) Use Mple to clculte integrls nd solve the liner system of equtions for i. Compre your solution with the exct solution sin πx u = π 2 (sinh πy + sinh(π(1 y) sinh π) sinh π by plotting u(x, y) s functions of x nd y for fixed vlues of y nd x, respectively. 42. The pproximte solution to the following problem u xx + u yy = 2, 1 <x<1, 1 <y<1 (1) u = on x = 1,x=1,y = 1,y =1 is sought by Kntorovich method with the following tril function u = f(x)(y 2 1). (2) Determine the form of the functionl tht leds to Eq. (1). Substitute (2) into this functionl nd perform integrtion over y. By vrition of the resulting functionl determine the Euler-Lgrnge eqution for f(x) nd solve it with pproprite boundry conditions. 9

10 43. Apply the lest squre 1 method to find the pproximte solution to the following nonliner eqution x + ωx 2 + εx 3 =, x() = A, x () =. Assume solution in the form u(t) =c cos(ωt) withω n unknown frequency. Find the frequency ω by minimizing the integrl R 2 (u)dt. Com- T pre the frequency with the vlue obtined by the method of verging for smll ε. 44. Consider eqution u u = exp(x)(exp( x)u ) = u() = 2 u(4) = 1 + e 4 Solve this eqution by method of finite elements using the vritionl form I = 4 exp(x)(u ) 2 dx. Use three elements, [, 1], [1, 2], nd [2, 4] nd employ qudrtic polynomils f i = α i + β i x + γ i x 2,i=1, 2, 3 Hint: Minimize I with the following constrints: boundry conditions (2 equtions), continuity of element functions nd derivtives t x=1 nd x=2 (4 equtions). Use Mple to determine α i,β i, nd γ i. Compre your solutions with the exct solution u(x) =1+e x. 1 In some textbooks this method is referred s the Ritz method 1