Transcript Slide 1

Mathematical Modeling and Simulation
Using
MATLAB
Differential Equations
(Plus Symbolic Mathematics)
Prof. Muhammad Saeed
1
§ Symbolic Math
syms x y z a b ;
1. Expansion:
expand((x+a)^3); expand(sin(a+b));
2. Factorization:
factor(a^2-b^2);
3. Series Summation:
syms k n;
symsum(k,0,10);symsum(k^2,1,4);sysmsum(k,0,n-1)
4. Substitution:
Expr=x^2+6*x+9; subs(Expr,x,2); subs(Expr,x,a);
5. Solution of Equations:
solve(‘3*x^3+2*x^2-4*x+12=0’);
solve(3*x^3+2*x^2-4*x+12);
solve(‘sin(2*x)-cos(x)=0’);
6.
Solution of Simultaneous Equations:
eqn1=‘4*x+3*y=5’; eqn2=‘2*x+2*y=-3’;
S=solve(eqn1,eqn2); { S.x, S.y}
S=solve(eqn1,eqn2, …….);
2
§ Symbolic Math:
7. Limit:
limit(sin(x)/x)
f = sin(x)/x;
limit(f); limit(f,a); limit(f, x,a,); limit(f,x,a,’right’);
limit(f,x,a,’left’);
8. Taylor Series:
f=exp(x); taylor(f, 5); taylor(f, 5,2);
9. Graph Plot:
ezplot(f, [-3 3])
10. Differentiation:
syms x n;
diff(x^n)
diff(‘(sin(x))^2’)
diff(x*sin(x*y), y, 2)
3
§ Symbolic Math:
11. Integration:
syms x y n a b
int(x^n);
int(x^n, n);
int(xy^2,y,0,4);
int(x^3, a,b);
12. Laplace Transform:
syms s b t w x;
laplace(t^3);
laplace(exp(-b*t)); laplace(exp(a*s)) ;
laplace(sin(w*x),t); laplace(cos(x*w),w,t);
laplace(diff(sym('F(t)')))
4
§ Symbolic Math:
13. Inverse Laplace Transform:
ilaplace(1/s^3);
ilaplace(1/(s+b);
ilaplace(b/(s^2+b^2));
14. More to study:
fourier, ifourier,
ztrans, iztrans,
sym,
poly2sym, sym2poly
findsym,
simplify,
collect
See for DE solution using Laplace Transform: DESymb.m
5
§ Symbolic Solution of Differential Equations:
∆
∆
∆
∆
∆
∆
∆
∆
dsolve(‘Dy+2*y = 12’)
dsolve(‘Dy = sin(a*t)’)
dsolve(‘D2y = c^2*y’)
[x, y] = dsolve(‘Dx = 3*x+4*y’, ‘Dy = -4*x+3*y’)
dsolve(‘Dy = sin(a*t)’, ‘y(0) = 0’) ; { ‘y(0) = c’ }
dsolve(‘D2y = c^2*y’,’y(0) = 1’,’Dy(0) = 0’)
[x, y] = dsolve(‘Dx = 3*x+4*y’, ‘Dy = -4*x+3*y’,’x(0) = 0’,’y(0) = 1’)
dsolve(‘D2y+9*sin(y) = 0’,’y(0) = 0’,’y(L) = 0’)
6
§ Numeric Solution of Differential Equations:
1. Euler’s Method.
2. MATLAB ODE Solvers
Example: DEEulersMethod.m
Solver
Solves These Kinds of Problems
Method
ode45
Nonstiff differential equations
Runge-Kutta
ode23
Nonstiff differential equations
Runge-Kutta
ode113
Nonstiff differential equations
Adams
ode15s
Stiff differential equations and DAEs
NDFs (BDFs)
ode23s
Stiff differential equations
Rosenbrock
ode23t
Moderately stiff differential equations and DAEs
Trapezoidal rule
ode23tb
Stiff differential equations
TR-BDF2
ode15i
Fully implicit differential equations
BDFs
7
§ Numeric Solution of Differential Equations:
a) DE’s Of Order 1:
[t, y]=solver(‘func’,[ti tj], y(i))
function ydot=funcName(t,y)
ydot= f(y,t)………. ;
end
ODE’s:
b) DE’s Of Order 2:
[t, x]=solver(‘function’,[ti, tj],[y(i), y(j)])
function xdot=funcName(t,x)
xdot(1)=x(2)
xdot(2)= func( x(1), x(2), t )
xdot=[xdot(1);xdot(2)];
end;
ODE:
,van der Pol Eqn.
8
§ Numeric Solution of Differential Equations:
Examples:
I. DE’s Of Order 1
function ydot = DEorder1_01(t,y)
ydot = sin(t);
end
[t,y] = ode23(‘DEorder1_01’, [0, 4*pi], 0);
Analytic Solution ‘ y= 1-cos(t) ‘
9
§ Numeric Solution of Differential Equations:
Examples:
2. DE’s Of Order 2
( van der Pol Eqn. )
function ydot = vdpol(t,y)
mu = 2;
ydot(1) = y(2);
ydot(2) = mu*(1-y(1)^2)*y(2) – y(1);
ydot = [ydot(1); ydot(2)];
end
[t,y] = ode45(‘vdpol’, [0, 20],[2; 0]);
10
§ Numeric Solution of Differential Equations:
Examples:
2. DE’s Of Order 2
( van der Pol Eqn. )
function ydot = vdpol2((t,y,mu)
ydot(1) = y(2);
ydot(2) = mu*(1-y(1)^2)*y(2) – y(1);
ydot = [ydot(1); ydot(2)]
end
options=odeset(‘Refine’,4);
[t,y] = ode45(‘vdpol’, [0, 20], [2; 0], options, 2);
11
END
12