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Q1/ What is the relationship between Laplace transform 00 L[u(t)] = F(s) = e-stu(t)dt 0. and the following transform -t E[u(t)] = T(w) = w ew u(t)dt Some Properties:- 1. E[1] = w² 2. E[t] = w³ 3. E[eax] = w2 1-aw Q2/ Use RDTM to solve the following nonlinear system: = u2v 2u+ +1 (4+1 Ju(x,y,t) at av(x,y,t) at with initial conditions 1/0²u =u-u²v+1 (+ 4x2 u(x, y, 0) = e-x-y v(x, y, 0) = ex+y J² u ay² a²u) ay2 Q3/ Find Adomian polynomial A6 for the nonlinear term F(u). Q4/Use homotopy perturbation method to solve the following Helmholtz equation: a²u(x, y) a²u(x, y) ax² + with initial conditions Jy2 - u(x, y) = 0, u(o,y) =y, ux(0,y) = y + coshy.

Q1/ What is the relationship between Laplace transform 00 L[u(t)] = F(s) = e-stu(t)dt 0. and the following transform -t E[u(t)] = T(w) = w ew u(t)dt Some Properties:- 1. E[1] = w² 2. E[t] = w³ 3. E[eax] = w2 1-aw Q2/ Use RDTM to solve the following nonlinear system: = u2v 2u+ +1 (4+1 Ju(x,y,t) at av(x,y,t) at with initial conditions 1/0²u =u-u²v+1 (+ 4x2 u(x, y, 0) = e-x-y v(x, y, 0) = ex+y J² u ay² a²u) ay2 Q3/ Find Adomian polynomial A6 for the nonlinear term F(u). Q4/Use homotopy perturbation method to solve the following Helmholtz equation: a²u(x, y) a²u(x, y) ax² + with initial conditions Jy2 - u(x, y) = 0, u(o,y) =y, ux(0,y) = y + coshy.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Q1/ What is the relationship between Laplace transform 00 L[u(t)] = F(s) = e-stu(t)dt 0. and the following transform -t E[u(t)] = T(w) = w ew u(t)dt Some Properties:- 1. E[1] = w² 2. E[t] = w³ 3. E[eax] = w2 1-aw Q2/ Use RDTM to solve the following nonlinear system: = u2v 2u+ +1 (4+1 Ju(x,y,t) at av(x,y,t) at with initial conditions 1/0²u =u-u²v+1 (+ 4x2 u(x, y, 0) = e-x-y v(x, y, 0) = ex+y J² u ay² a²u) ay2 Q3/ Find Adomian polynomial A6 for the nonlinear term F(u). Q4/Use homotopy perturbation method to solve the following Helmholtz equation: a²u(x, y) a²u(x, y) ax² + with initial conditions Jy2 - u(x, y) = 0, u(o,y) =y, ux(0,y) = y + coshy.
Transcribed Image Text:Q1/ What is the relationship between Laplace transform 00 L[u(t)] = F(s) = e-stu(t)dt 0. and the following transform -t E[u(t)] = T(w) = w ew u(t)dt Some Properties:- 1. E[1] = w² 2. E[t] = w³ 3. E[eax] = w2 1-aw Q2/ Use RDTM to solve the following nonlinear system: = u2v 2u+ +1 (4+1 Ju(x,y,t) at av(x,y,t) at with initial conditions 1/0²u =u-u²v+1 (+ 4x2 u(x, y, 0) = e-x-y v(x, y, 0) = ex+y J² u ay² a²u) ay2 Q3/ Find Adomian polynomial A6 for the nonlinear term F(u). Q4/Use homotopy perturbation method to solve the following Helmholtz equation: a²u(x, y) a²u(x, y) ax² + with initial conditions Jy2 - u(x, y) = 0, u(o,y) =y, ux(0,y) = y + coshy.
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