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1 gnuplot gnuplot dy dx = f(x, y) y(x) = x x 0 f(x, y)dx + y(x 0 )

2 dy dx y(x + h) y(x) h y(x + h) y(x) h = f(x, y(x)) y(x + h) = y(x) + hf(x, y(x)) dy dx y(x + h) y(x h) 2h y(x + h) = y(x h) + 2hf(x, y(x)) y(x) y(x + h) y(x + h) y(x) y(x h) y(x + h) y(x + h) = y(x) + h dy dx + h2 d 2 y h2 + = y(x) + hf(x, y) + 2! dx2 2 f (x, y) + h3 6 f (x, y) + h 2 f(x, y) + 2 h // euler method 2 // y = xy ==> log(y) = xˆ2/2, y = exp(xˆ2/2); 3 #include "stdafx.h" 4 #include <math.h> 5 6 double f(double x, double y); 7 8 int 9 main(int ac, char av[])

3 { 11 double x, y, h, s; 12 double r, e; 13 int i; h = 0.1; 16 x = 0; 17 y = 1; 18 printf("x=%8.5f, y=%8.5f\n", x, y); 19 for(i = 0; i < 20; i++) { 20 s = f(x, y); 21 y = y + s h; 22 x = x + h; 23 r = exp(x x/2); 24 e = (y r)/r; 25 printf("x=%8.5f, y=%8.5f, r=%8.5f, e = %8.5f\n", x, y, r, e); 26 } 27 } double f(double x, double y) 30 { 31 double r; 32 r = x y; 33 return r; 34 } 7.2 y(x + h) y(x + h) = y(x) + hf(x, y) + h2 2 f (x, y) + h3 6 f (x, y) + f(x, y) f(x, y) + h/2f (x, y) + h 2 /6f (x, y) 3 1 // taylor method 2 // y = xy ==> log(y) = xˆ2/2, y = exp(xˆ2/2); 3 #include "stdafx.h" 4 #include <math.h> 5 6 double taylor(double x, double y, double h); 7 8 int 9 main(int ac, char av[]) 10 { 11 double x, y, h, s; 12 double r, e;

4 int i; h = 0.1; 16 x = 0; 17 y = 1; 18 printf("x=%8.5f, y=%8.5f\n", x, y); 19 for(i = 0; i < 20; i++) { 20 s = taylor(x, y, h); 21 y = y + s h; 22 x = x + h; 23 r = exp(x x/2); 24 e = (y r)/r; 25 printf("x=%8.5f, y=%8.5f, r=%8.5f, e = %8.5f\n", x, y, r, e); 26 } 27 } double taylor(double x, double y, double h) 30 { 31 double r1,r2,r3; 32 r1 = x y; // f 33 r2 = y + x r1; // f = y + xy = y+xf 34 r3 = 2 r1 + x r2; // f = 2y + xf = 2f+xf 35 return r1+h (r2/2+h r3/6); 36 } f(x, y) f(x, y) = xy f(x, y) df/dx = (f(x + h) f(x))/h d f f(x, y) = dx x + dy f dx y = f x + f f y 1 double taylor(double x, double y, double h) 2 { 3 double r1,r2,r3; 4 double fx, fy, fxx, fxy, fyy; 5 r1 = f(x,y); // f(x, y); 6 fx = f(x+h, y) r1; // f(x+h, y) f(x,y); 7 fy = f(x, y+h) r1; // f(x, y+h) f(x,y); 8 fxx = fx + f(x h, y) r1; // f(x+h, y) + f(x h, y) 2f(x, y); 9 fyy = fy + f(x, y h) r1; // f(x, y+h) + f(x, y h) 2f(x, y) 10 fxy = f(x+h, y+h) fx fy r1; // f(x+h, y+h) f(x, y+h) f(x+h, y) + f(x, y); 11 r2 = fx+r1 fy; // f = f x + f f y 12 r3 = fxx+2 r1 fxy+fx fy+r1 r1 fyy; // f = f xx +2f f xy + f x f y + fˆ2 f yy 13 return r1+r2/2+r3/6; 14 } 15

5 double f(double x, double y) 17 { 18 double r; 19 r = x y; // f 20 return r; 21 } 7.3 y(x + h) = y(x) + hf(x, y) + h2 2 f (x, y) + h3 6 f (x, y) + f (x, y) f (x, y) = f(x, y) + x f (x, y) = 2 f(x, y) f(x, y) f(x, y) y f(x, y) x 2 x y f(x, y) + 2 f(x, y) f 2 (x, y) + y 2 ( ) 2 f(x, y) + f(x, y) y f(x, y) f(x, y) f(x + s, y + t) = f(x, y) + s + t x y + s2 2 2 f(x, y) + st 2 f(x, y) + t2 x 2 x y 2 f(x, y) f(x, y) y x 2 f(x, y) y 2 + hf + h2 2 (f x + f y f) (1) h{(1 α)f(x, y) + αf(x + s, y + t)} hf + αhsf x + αhtf y (2) h/2 = αs hf/2 = αt s = h, t = h f (1) (2) 2α 2α y(x + h) = y(x) + h((1 α)f(x, y) + αf(x + h 2α, y + h f(x, y)) 2α α = 1/2 y(x + h) = y(x) + h(f(x, y) + f(x + h, y + hf(x, y))

6 k 1 = f(x, y) k 2 = f(x + s 1, y + t 11 k 1 ) k 3 = f(x + s 2, y + t 21 k 1 + t 22 t 2 ). k n = f(x + s n, y + t n1 k 1 + t n2 k t nn 1 k n 1 ) y(x + h) = y(x) + h{α 1 k 1 + α 2 k α n k n } n 4 k 1 = hf(x, y) k 2 = hf(x + h, y + k ) 2 ) k 3 = hf(x + h, y + k 2 2 k 4 = hf(x + h, y + k 3 ) y(x + h) = y(x) (k 1 + 2k 2 + 2k 3 + k 4 ) y(x + h) f(x, y) x // runge kutta method 2 // y = xy ==> log(y) = xˆ2/2, y = exp(xˆ2/2); 3 #include "stdafx.h" 4 #include <math.h> 5 6 double f(double x, double y); 7 8 int 9 main(int ac, char av[]) 10 { 11 double x, y, h; 12 double k1, k2; 13 double r, e; 14 int i; h = 0.1; 17 x = 0; 18 y = 1;

7 printf("x=%8.5f, y=%8.5f\n", x, y); 20 for(i = 0; i < 20; i++) { 21 k1 = f(x, y); 22 k2 = f(x+h, y+ k1 h); 23 y = y + h (k1 + k2); 24 x = x + h; 25 r = exp(x x/2); 26 e = (y r)/r; 27 printf("x=%8.5f, y=%8.5f, r=%8.5f, e = %8.5f\n", x, y, r, e); 28 } 29 } double f(double x, double y) 32 { 33 double r; 34 r = x y; 35 return r; 36 } // 2nd order runge kutta 2 #include "stdafx.h" 3 #include <math.h> 4 5 #define N 2 6 #define M void func(double x, double y[], double f[]); 9 10 int 11 main(int ac, char av[]) 12 { double x, h; 15 int i, j; 16 double y[n], v[n], f[n]; 17 double k[2][n]; 18 FILE fo; fo = fopen("vol_2.dat", "w"); 21 h = 0.1; 22 x = 0; 23 y[0] = 0.1; 24 y[1] = 0.1; for (i = 0; i < M; i++) {

8 printf("x=%8.5f, y=%8.5f, %8.5f\n", x, y[0], y[1]); 28 fprintf(fo, "%8.5f %8.5f %8.5f\n", x, y[0], y[1]); 29 func(x, y, f); 30 for (j = 0; j < N; j++) 31 k[0][j] = h f[j]; 32 for (j = 0; j < N; j++) 33 v[j] = y[j] + k[0][j]; 34 func(x + h, v, f); 35 for (j = 0; j < N; j++) 36 k[1][j] = h f[j]; 37 x = x + h; 38 for (j = 0; j < N; j++) { 39 y[j] = y[j] + (k[0][j] + k[1][j]); 40 } 41 } 42 fclose(fo); 43 } void 46 func(double x, double y[n], double f[n]) 47 { 48 f[1] = (1 y[0] y[0]) y[1] y[0]; // dz/dx = (1 y y) z y 49 f[0] = y[1]; // dy/dx = z 50 } // 4th order runge kutta 2 #include "stdafx.h" 3 #include <math.h> 4 5 #define N 2 6 #define M void func(double x, double y[], double f[]); 9 10 int 11 main(int ac, char av[]) 12 { double x, h; 15 int i, j; 16 double y[n], v[n], f[n]; 17 double k[4][n]; 18 FILE fo; fo = fopen("vol_4.dat", "w");

9 h = 0.1; 22 x = 0; 23 y[0] = 0.1; 24 y[1] = 0.1; for (i = 0; i < M; i++) { 27 printf("x=%8.5f, y=%8.5f, %8.5f\n", x, y[0], y[1]); 28 fprintf(fo, "%8.5f %8.5f %8.5f\n", x, y[0], y[1]); 29 func(x, y, f); 30 for (j = 0; j < N; j++) 31 k[0][j] = h f[j]; 32 for (j = 0; j < N; j++) 33 v[j] = y[j] + k[0][j] / 2; 34 func(x + h / 2, v, f); 35 for (j = 0; j < N; j++) 36 k[1][j] = h f[j]; 37 for (j = 0; j < N; j++) 38 v[j] = y[j] + k[1][j] / 2; 39 func(x + h / 2, v, f); 40 for (j = 0; j < N; j++) 41 k[2][j] = h f[j]; 42 for (j = 0; j < N; j++) 43 v[j] = y[j] + k[2][j]; 44 func(x + h, v, f); 45 for (j = 0; j < N; j++) 46 k[3][j] = h f[j]; 47 x = x + h; 48 for (j = 0; j < N; j++) { 49 y[j] = 50 y[j] + (k[0][j] + 2 k[1][j] + 2 k[2][j] + k[3][j]) / 6; 51 } 52 } 53 fclose(fo); 54 } void 57 func(double x, double y[n], double f[n]) 58 { 59 f[0] = (1 y[1]) y[0]; // dy/dx= (1 z)y 60 f[1] = 2 (1 y[0]) y[1]; // dz/dx= 2(1 y)z 61 } 7.4 x y x, y

10 7. 10 dx dt dy dt = (y 1)x = 2(1 x)y x(t = 0) = y(t = 0) = vol-1.dat 7 vol-2.dat dy dx = ay y(x) = y(0)e ax y n+1 y n h = ay n y n+1 = (1 ah)y n y n y n = (1 ah) n y 0 y n+1 y n 1 2h = ay n

11 7. 11 y n+1 + 2ahy n y n 1 = 0 t 2 + 2aht 1 = 0 t 1,2 = ah ± a 2 h y n = At n 1 + Bt n 2 a > 0 t 2 = ( a 2 h ah) < 1 t n 2 n y 0 = y 0, y 1 = y 0 e ah A, B A = 1 ( ) 1 + e ah + ah y 0 2 a2 h B = 1 ( ) 1 e ah + ah y 0 2 a2 h B n y n Bt n dy dx dz dx y(0) = 0.5, z(0) = 0.1 y(x), z(x) = (2 z)y = (3 2y)z 7.2 d 2 y dx µ(1 2 y2 ) dy dx + y = 0 y(0) = dy (0) = 1 µ = 1 dx µ µ = 100 y(x) n d k dx k y = y i, i = 0,..., n 1

12 7. 12 d y dx n 1 = F (y n 1,..., y 0, x) d y dx n 2 = y n 1. d y dx 0 = y 1 { d dx y 1 = µ(1 y 2 0)y 1 y 0 d dx y 0 = y dx dt dy dt dz dt = 10x + 10y = 28x y xz = 8 3 z + xy 4 x(0) = 0, y(0) = 1, z(0) = 10 h = 0.01 t = 0 t = 100 x, y, z 3 gnuplot splot with line

13 gnuplot gnuplot gnuplot https : //sourcef orge.net/projects/gnuplot/ wgnuplot gp***-win32-setup.exe (gp***- win32-mingw.zip gp***-win64-mingw.zip) OS 32 win32 64 win64 gnuplot gnuplot bin wgnuplot.ext wgnuplot.exe wgnuplot terminal type set "wxt" gnuplot > > sin(x) terminal type set "wxt" gnuplot > plot sin(x) sin

14 7. 14 x y (sample.dat wgnuplot terminal type set "wxt" gnuplot > plot "sample.dat" cd

15 7. 15 gnuplot plot x**3 + x + 1 plot sin(x), cos(x) plot [ 0:6.28] [-1.5:1.5] sin(x) splot x**2+y**2 plot sample.dat plot sample.dat with line plot sample.dat with linepoints plot sample.dat using 1:2 splot sample.dat using 2:3:4 with line x 3 + x + 1 sin(x) 0 x 2π, 1.5 y 1.5 x 2 + y 2 sample.dat sample.dat sample.dat x 2 y x, y, z 2,3,4 x, y, z gif terminal ( ) gif gnuplot > set terminal "gif" gnuplot >set output "sample.gif" gnuplot > plot "sample.dat" with line sample.gif gif gif

16 7. 16 gnuplot > set terminal "wxt"

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