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Problem 1: Consider the following non-linear, implicit integral equation u(x) = exp [where t k (Ix - x'l) u(x') dX'] k(lxl) = t exp( -Ixl) Discretize...

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Problem 1: Consider the following non-linear, implicit integral equation u(x) = exp [- t k (Ix - x'l) u(x') dX'] where k(lxl) = t exp( -Ixl) Discretize the above equation by defining Uj = u(Xj), Xi = (i-I )L\x, i = I, 2, . .. n = 41, and use trapezoidal rule to evaluate the integral. (a) Solve the resulting equation by the method of successive substitutions. Accept as a solution the value of Ujat the k th iteration for which (b) Solve the same problem using the Newton-Raphson method. Note that numerical integration by Trapezoidal rule is given by b [ n-I ] !f(x)dx= ~ f(a)+2 i~2f(Xi)+ feb) , where ~= (:-=-~) Hint: eBE 9111 (Ray, AK)
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where lex) = 1 texp[- ~x - x'l)u(x') ca']= 1 f(x,x')ca' o 0 f(x,x') = texp{-Ix - x'i )u(x') For successive substitution method u?) = u(x?)) = ex p[ - l(x?-I))] for i = 1,2, .. ,41, and k = 0, 1,2, .. where For Newton-Raphson method Define and solve J ~u = - g Problem 2 The following set of equations describes the dynamic behavior of a fluidized bed reactor dYI 4 dt = 1.30(Y3 - YI) + 1.04 x 10 ky2 d~; = 1.88 x 10 3 [y 4 - Y2 (1 + k)] dY3 "dt = 1752 + 266.7 YI - 269.3Y3 dY4 d/=0.1+320Y2 -321Y4 where k = 0.0006exp[ 20.7 - 15: 1 00]. Yl(O) = 759.167 Y2(0) = 0.0 Y3(0) = 600.0 YI(O) = 0.1 CBE 9111 (Ray, AK) 2
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