Perturbation Methods [PDF]

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Q.1



(a) Explain order symbol and gauge functions with examples (only definitions). (b) For small  , determine two terms in the expansion of each root of the following equation



 x 2  x  1  0. Q.2



(a) Find the two terms perturbation solution of the following initial value problem:



df  f   f 2, dt



f (0)  1,



(0 <   1).



(b) Find the sources of non-uniformity in the following problems, (i )  y / /  y /  y  0, y (0)  a, y (1)  b,



Q. 3



(ii ) ( x  y )



dy  (2  x) y  0, y (1)  e 1 dx



(a) Use Adomian decomposition method to solve the following inhomogeneous PDE



u x  u y  x  y,



u (0, y)  0,



u ( x,0)  0.



Check whether x -solution and y -solutions match or not? (b) Without solving the following equation, explain why Adomian Decomposition Method is not applicable?



ux  u y  0, Q.4



u (0, y)  0,



u ( x, 0)  0.



(a) Explain the sources of non-uniformity when applying a straightforward expansion in terms of a small parameter for nonlinear problems. (b) Solve the following partial differential equation



u x  u x  ux  u, Q. 5



u (0, y, z )  1  e y  e z , u ( x, 0, z )  1  e x  e z , u ( x, y, 0)  1  e y  e x .



(a) Explain the phenomena of the “Noise terms” with details and examples. (b) Use the decomposition method and the noise terms phenomenon to solve the following PDE



ux  yu y  y (cosh x  sinh x), Q. 6



u (0, y )  y,



u ( x, 0)  0.



(a) Calculate the Adomian Polynomials for k  0,1, 2 for the following functions,



(i) F (u )  e2u (ii) F (u )  u 2 u x (iii ) F (u )  ln u (b) Use Adomian decomposition method to solve the second order nonlinear differential equation



y / /  ( y / ) 2  y 2  1- sin x,



y (0)  0, y / (0)  1.