Madhukesh J K

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These are questions asked by Madhukesh J K

Dear all
Warm Greetings.

I want to display a solution (only numerical value) of the first derivative of the function for the values of x varies from 0 to 7 with step size 0.01.

I have attached the work file. 
ODE.mw

Please do the needful.

Thanks in advance.

I am getting errors while applying the HPM method.
I have attached the maple file.

HPM.mw

Please help me to get the solution.

Thanks in advance.

I am trying to obtain the solution of the differential equation f'''+ff''-f'^2-Mf'=0, with f(0)=0, f'(0)=1, and f'(5)=0 with M=0.5 using finite element method

But got this error. I attached the file also.

restart

with(LinearAlgebra):

with(plots):

M := .5;

.5

(1)

a := 0;

0

(2)

b := 5;

5

(3)

N := 50;

50

(4)

h := (b-a)/N;

1/10

(5)

nodes := [seq(h*i+a, i = 0 .. N)];

[0, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, 1, 11/10, 6/5, 13/10, 7/5, 3/2, 8/5, 17/10, 9/5, 19/10, 2, 21/10, 11/5, 23/10, 12/5, 5/2, 13/5, 27/10, 14/5, 29/10, 3, 31/10, 16/5, 33/10, 17/5, 7/2, 18/5, 37/10, 19/5, 39/10, 4, 41/10, 21/5, 43/10, 22/5, 9/2, 23/5, 47/10, 24/5, 49/10, 5]

(6)

elements := [seq([i, i+1], i = 0 .. N-1)];

[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22], [22, 23], [23, 24], [24, 25], [25, 26], [26, 27], [27, 28], [28, 29], [29, 30], [30, 31], [31, 32], [32, 33], [33, 34], [34, 35], [35, 36], [36, 37], [37, 38], [38, 39], [39, 40], [40, 41], [41, 42], [42, 43], [43, 44], [44, 45], [45, 46], [46, 47], [47, 48], [48, 49], [49, 50]]

(7)

bilinear := proc (u, v, w) options operator, arrow; int(diff(u(x), `$`(x, 3))+u(x)*(diff(u(x), `$`(x, 2)))-(diff(u(x), x))^2-M*u(x)*(diff(u(x), x)), x = w[1] .. w[2])+int((diff(u(x), x))*(diff(v(x), x)), x = w[1] .. w[2]) end proc;

proc (u, v, w) options operator, arrow; int(diff(u(x), `$`(x, 3))+u(x)*(diff(u(x), `$`(x, 2)))-(diff(u(x), x))^2-M*u(x)*(diff(u(x), x)), x = w[1] .. w[2])+int((diff(u(x), x))*(diff(v(x), x)), x = w[1] .. w[2]) end proc

(8)

Llinear := proc (v, w) options operator, arrow; v(a)*(diff(w(x), x)) end proc, x = a;

proc (v, w) options operator, arrow; v(a)*(diff(w(x), x)) end proc, x = 0

(9)

K := CreateMatrix(N+1, N+1, 0);

CreateMatrix(51, 51, 0)

(10)

F := CreateVector(N+1, 0);

CreateVector(51, 0)

(11)

for e in elements do x1 := nodes[e[1]]; x2 := nodes[e[2]]; h := x2-x1; Ke := bilinear(proc (x) options operator, arrow; piecewise(x < x1+(1/2)*h, 1-(x-x1)/h, (x2-x)/h) end proc, proc (x) options operator, arrow; piecewise(x < x1+(1/2)*h, (x-x1)/h, (x2-x)/h) end proc, [x1, x2]); Fe := Llinear(proc (v) options operator, arrow; v(x)*piecewise(x = x1, 1, x <> x1) end proc, [x1, x2]); for i in [e[1], e[2]] do for j in [e[1], e[2]] do K[i, j] := K[i, j]+Ke[i-e[1]+1, j-e[1]+1] end do; F[i] := F[i]+Fe[i-e[1]+1] end do end do

Error, invalid subscript selector

 

K[1, 1] := 1;

1

 

0

 

0

(12)

K[N+1, N+1] := 1;

1

 

0

(13)

u := LinearSolve(K, F)

Error, (in LinearAlgebra:-LinearSolve) invalid input: LinearAlgebra:-LinearSolve expects its 1st argument, A, to be of type {Matrix, list({Matrix, Vector})} but received K

 

f := unapply(u(x), x);

proc (x) options operator, arrow; u(x) end proc

 

Warning, unable to evaluate the function to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct

 

 

``

``

Download FEM.mw

How to integrate the below function from 0 to eta.

A := P(eta)+S(H-2*`cos&theta;`(eta+1)*F)+`cos&theta;`*(2*F(eta)-2*H*F) = S(eta+1)*`sin&theta;`*F(theta)-`sin&theta;`(F*H(theta)-H*F(theta)+F(theta, eta))

Thanks in advance

HI, I have numerically solved the given problem using the dsolve command But I want to solve the same problem using the Differential transformation method.
Can anyone help me to get the series solution for the given problem using DTM.

I want to compare the numerical results with DTM results when lambda =0.5.

eqn1 := diff(f(eta), `$`(eta, 3))+f(eta)*(diff(f(eta), `$`(eta, 2)))-(diff(f(eta), eta))^2-lambda*(diff(f(eta), eta)) = 0.

eqn2 := diff(theta(eta), `$`(eta, 2))+f(eta)*(diff(theta(eta), eta))*Pr = 0

Bcs := (D(f))(0) = 1, f(0) = 0, (D(f))(infinity) = 0, theta(0) = 1, theta(infinity) = 0;

[lambda = .5, Pr = 6.3]

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