-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathcahnAllen_mF.txt
162 lines (139 loc) · 10.1 KB
/
cahnAllen_mF.txt
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
==========Modified F-Expansion Method===========
Equations are written in GiNaC language.
----------------------------------------------------------------------------------------------------
Input equation is: -u-Diff(u,x,2)+Diff(u,t,1)+u^3 = 0;
The Diff. Equ. becomes: Diff(U,xi,1)*k_0+U^3-U-k_1^2*Diff(U,xi,2) = 0;
u = U,
where xi = k_1*x+t*k_0;
The value of N is: 1;
U = a_1*F+a_0;
The first-order nonlinear ODE: diff(F,xi,1) = A_2*F^2+A_1*F;
****************************************************************************************************
The system of algebraic equations are:
F^0: -a_0+a_0^3 = 0;
F^1: 3*a_1*a_0^2-a_1+a_1*A_1*k_0-a_1*k_1^2*A_1^2 = 0;
F^2: 3*a_1^2*a_0-3*a_1*k_1^2*A_2*A_1+a_1*A_2*k_0 = 0;
F^3: a_1^3-2*a_1*k_1^2*A_2^2 = 0;
In the following results C_ is an arbitrary constant.
****************************************************************************************************
solving above system of equations for variables {k_0,k_1,a_0,a_1}->
{k_0==-3/2*A_1^(-1),a_0==-1,a_1==-A_2*A_1^(-1),k_1==1/2*sqrt(2)*A_1^(-1)}
U = -1-A_2*A_1^(-1)*F,
where F is the solution of
diff(F,xi,1) = A_2*F^2+A_1*F;
solution(s) of input Diff. Equ. is (are)=>
solution #1 u = -1+(cosh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)+sinh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1))*(-1+cosh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)*A_2+sinh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)*A_2)^(-1)*A_2;
solution #2 u = -1-A_2*A_1^(-1)*(exp(-1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)*C_-A_2*A_1^(-1))^(-1);
solution #3 u = -1+1/2*(tanh(1/4*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
solution #4 u = -1+1/2*(A_2^(-1)*A_1+coth(1/4*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1)*A_2*A_1^(-1);
{k_0==3/2*A_1^(-1),a_0==0,a_1==-A_2*A_1^(-1),k_1==1/2*sqrt(2)*A_1^(-1)}
U = -A_2*A_1^(-1)*F,
where F is the solution of
diff(F,xi,1) = A_2*F^2+A_1*F;
solution(s) of input Diff. Equ. is (are)=>
solution #5 u = (cosh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)+sinh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1))*(-1+cosh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)*A_2+sinh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)*A_2)^(-1)*A_2;
solution #6 u = -A_2*A_1^(-1)*(exp(-1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)*C_-A_2*A_1^(-1))^(-1);
solution #7 u = 1/2*(tanh(1/4*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
solution #8 u = 1/2*(A_2^(-1)*A_1+coth(1/4*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1)*A_2*A_1^(-1);
{k_1==sqrt(2)*A_1^(-1),k_0==0,a_0==-1,a_1==-2*A_2*A_1^(-1)}
U = -1-2*A_2*A_1^(-1)*F,
where F is the solution of
diff(F,xi,1) = A_2*F^2+A_1*F;
solution(s) of input Diff. Equ. is (are)=>
solution #9 u = -1+2*(sinh(sqrt(2)*x+A_1*C_)+cosh(sqrt(2)*x+A_1*C_))*A_2*(-1+sinh(sqrt(2)*x+A_1*C_)*A_2+cosh(sqrt(2)*x+A_1*C_)*A_2)^(-1);
solution #10 u = -1-2*(exp(-sqrt(2)*x)*C_-A_2*A_1^(-1))^(-1)*A_2*A_1^(-1);
solution #11 u = -1+(tanh(1/2*sqrt(2)*x+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
solution #12 u = -1+(A_2^(-1)*A_1+A_2^(-1)*coth(1/2*sqrt(2)*x+C_)*A_1)*A_2*A_1^(-1);
{k_1==sqrt(2)*A_1^(-1),k_0==0,a_1==2*A_2*A_1^(-1),a_0==1}
U = 1+2*A_2*A_1^(-1)*F,
where F is the solution of
diff(F,xi,1) = A_2*F^2+A_1*F;
solution(s) of input Diff. Equ. is (are)=>
solution #13 u = 1-2*(sinh(sqrt(2)*x+A_1*C_)+cosh(sqrt(2)*x+A_1*C_))*A_2*(-1+sinh(sqrt(2)*x+A_1*C_)*A_2+cosh(sqrt(2)*x+A_1*C_)*A_2)^(-1);
solution #14 u = 1+2*(exp(-sqrt(2)*x)*C_-A_2*A_1^(-1))^(-1)*A_2*A_1^(-1);
solution #15 u = 1-(tanh(1/2*sqrt(2)*x+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
solution #16 u = 1-(A_2^(-1)*A_1+A_2^(-1)*coth(1/2*sqrt(2)*x+C_)*A_1)*A_2*A_1^(-1);
{a_0==-1,a_1==0}
solution(s) of input Diff. Equ. is (are)=>
solution #17 u = -1;
{k_0==3/2*A_1^(-1),a_1==A_2*A_1^(-1),a_0==0,k_1==-1/2*sqrt(2)*A_1^(-1)}
U = A_2*A_1^(-1)*F,
where F is the solution of
diff(F,xi,1) = A_2*F^2+A_1*F;
solution(s) of input Diff. Equ. is (are)=>
solution #18 u = -(-1+cosh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)*A_2+sinh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)*A_2)^(-1)*(sinh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)+cosh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1))*A_2;
solution #19 u = A_2*A_1^(-1)*(exp(1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)*C_-A_2*A_1^(-1))^(-1);
solution #20 u = -1/2*(tanh(-1/4*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
solution #21 u = -1/2*(A_2^(-1)*A_1+coth(-1/4*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1)*A_2*A_1^(-1);
{k_0==-3/2*A_1^(-1),k_1==-1/2*sqrt(2)*A_1^(-1),a_0==-1,a_1==-A_2*A_1^(-1)}
U = -1-A_2*A_1^(-1)*F,
where F is the solution of
diff(F,xi,1) = A_2*F^2+A_1*F;
solution(s) of input Diff. Equ. is (are)=>
solution #22 u = -1+(-1+cosh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)*A_2+sinh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)*A_2)^(-1)*(sinh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)+cosh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1))*A_2;
solution #23 u = -1-A_2*A_1^(-1)*(exp(1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)*C_-A_2*A_1^(-1))^(-1);
solution #24 u = -1+1/2*(tanh(-1/4*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
solution #25 u = -1+1/2*(A_2^(-1)*A_1+coth(-1/4*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1)*A_2*A_1^(-1);
{k_0==0,a_0==-1,a_1==-2*A_2*A_1^(-1),k_1==-sqrt(2)*A_1^(-1)}
U = -1-2*A_2*A_1^(-1)*F,
where F is the solution of
diff(F,xi,1) = A_2*F^2+A_1*F;
solution(s) of input Diff. Equ. is (are)=>
solution #26 u = -1+2*(sinh(-sqrt(2)*x+A_1*C_)+cosh(-sqrt(2)*x+A_1*C_))*A_2*(-1+sinh(-sqrt(2)*x+A_1*C_)*A_2+A_2*cosh(-sqrt(2)*x+A_1*C_))^(-1);
solution #27 u = -1-2*(exp(sqrt(2)*x)*C_-A_2*A_1^(-1))^(-1)*A_2*A_1^(-1);
solution #28 u = -1+(tanh(-1/2*sqrt(2)*x+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
solution #29 u = -1+(coth(-1/2*sqrt(2)*x+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
{k_0==0,a_1==2*A_2*A_1^(-1),a_0==1,k_1==-sqrt(2)*A_1^(-1)}
U = 1+2*A_2*A_1^(-1)*F,
where F is the solution of
diff(F,xi,1) = A_2*F^2+A_1*F;
solution(s) of input Diff. Equ. is (are)=>
solution #30 u = 1-2*(sinh(-sqrt(2)*x+A_1*C_)+cosh(-sqrt(2)*x+A_1*C_))*A_2*(-1+sinh(-sqrt(2)*x+A_1*C_)*A_2+A_2*cosh(-sqrt(2)*x+A_1*C_))^(-1);
solution #31 u = 1+2*(exp(sqrt(2)*x)*C_-A_2*A_1^(-1))^(-1)*A_2*A_1^(-1);
solution #32 u = 1-(tanh(-1/2*sqrt(2)*x+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
solution #33 u = 1-(coth(-1/2*sqrt(2)*x+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
{k_0==-3/2*A_1^(-1),a_1==A_2*A_1^(-1),a_0==1,k_1==-1/2*sqrt(2)*A_1^(-1)}
U = 1+A_2*A_1^(-1)*F,
where F is the solution of
diff(F,xi,1) = A_2*F^2+A_1*F;
solution(s) of input Diff. Equ. is (are)=>
solution #34 u = 1-(-1+cosh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)*A_2+sinh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)*A_2)^(-1)*(sinh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)+cosh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1))*A_2;
solution #35 u = 1+A_2*A_1^(-1)*(exp(1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)*C_-A_2*A_1^(-1))^(-1);
solution #36 u = 1-1/2*(tanh(-1/4*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
solution #37 u = 1-1/2*(A_2^(-1)*A_1+coth(-1/4*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1)*A_2*A_1^(-1);
{k_0==3/2*A_1^(-1),a_1==A_2*A_1^(-1),a_0==0,k_1==1/2*sqrt(2)*A_1^(-1)}
U = A_2*A_1^(-1)*F,
where F is the solution of
diff(F,xi,1) = A_2*F^2+A_1*F;
solution(s) of input Diff. Equ. is (are)=>
solution #38 u = -(cosh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)+sinh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1))*(-1+cosh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)*A_2+sinh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)*A_2)^(-1)*A_2;
solution #39 u = A_2*A_1^(-1)*(exp(-1/2*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1)*C_-A_2*A_1^(-1))^(-1);
solution #40 u = -1/2*(tanh(1/4*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
solution #41 u = -1/2*(A_2^(-1)*A_1+coth(1/4*(sqrt(2)*A_1^(-1)*x+3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1)*A_2*A_1^(-1);
{a_0==0,a_1==0}
solution(s) of input Diff. Equ. is (are)=>
solution #42 u = 0;
{k_0==-3/2*A_1^(-1),a_1==A_2*A_1^(-1),a_0==1,k_1==1/2*sqrt(2)*A_1^(-1)}
U = 1+A_2*A_1^(-1)*F,
where F is the solution of
diff(F,xi,1) = A_2*F^2+A_1*F;
solution(s) of input Diff. Equ. is (are)=>
solution #43 u = 1-(cosh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)+sinh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1))*(-1+cosh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)*A_2+sinh(A_1*C_+1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)*A_2)^(-1)*A_2;
solution #44 u = 1+A_2*A_1^(-1)*(exp(-1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)*C_-A_2*A_1^(-1))^(-1);
solution #45 u = 1-1/2*(tanh(1/4*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
solution #46 u = 1-1/2*(A_2^(-1)*A_1+coth(1/4*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1)*A_2*A_1^(-1);
{k_0==3/2*A_1^(-1),a_0==0,k_1==-1/2*sqrt(2)*A_1^(-1),a_1==-A_2*A_1^(-1)}
U = -A_2*A_1^(-1)*F,
where F is the solution of
diff(F,xi,1) = A_2*F^2+A_1*F;
solution(s) of input Diff. Equ. is (are)=>
solution #47 u = (-1+cosh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)*A_2+sinh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)*A_2)^(-1)*(sinh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)+cosh(A_1*C_-1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1))*A_2;
solution #48 u = -A_2*A_1^(-1)*(exp(1/2*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1)*C_-A_2*A_1^(-1))^(-1);
solution #49 u = 1/2*(tanh(-1/4*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1+A_2^(-1)*A_1)*A_2*A_1^(-1);
solution #50 u = 1/2*(A_2^(-1)*A_1+coth(-1/4*(sqrt(2)*A_1^(-1)*x-3*A_1^(-1)*t)*A_1+C_)*A_2^(-1)*A_1)*A_2*A_1^(-1);
{a_0==1,a_1==0}
solution(s) of input Diff. Equ. is (are)=>
solution #51 u = 1;
****************************************************************************************************
****************************************************************************************************
Time: 1.525 seconds