Typically: is given. Also given initial conditions like at
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| Solving Ordinary differential Equations (ODE)
Typically:  is given. Also given initial conditions like at  .
Typical examples: Equation 1: 
Equation 2: Equation 3: Equation 4: Equation 5: Solution 1: from The solution: 
To verify using MATLAB: First save this function as an .m file in MATLAB work space. function dy = g1(x,y) % This function evaluates the first-order % ODE usign Runge-Kutta method. % dy stands for the derivative of y % % dy = 3.0*x.^2; Next we go tot eh command window and develop the solution there. EDU» clear EDU» % Obtain solution of ODE1 EDU» [x,num_y]=ode23('g1',2,4,0.5); EDU» y=x.^3 -7.5; EDU» hndl1=plot(x,num_y)
EDU» hold on EDU» hndl2=plot(x,y,'o')
EDU»
Another one. This time the function is more complicated one. function dy = g2(x,y) dy = 3*y + exp(2*x); The domain and initial conditions are: From EDU» clear EDU» clear EDU» [x,num_y]=ode23('g2',0,3,3); EDU» hndl1=plot(x,num_y) hndl1 = 1.0018 EDU» set(hndl1,'LineWidth',3) EDU»
EDU» %
EDU» % We now attempt ode45 instead EDU» %
EDU» hold on EDU» [x,y1]=ode45('g2',0,3,3); EDU» hndl2=plot(x,y1,'o')
EDU»
An Ode23 estimates the function by 2nd order and 3rd order Taylor series, i.e. using, say, three consecutive near-by tangents to the function. Here, the functional form of the differential wouldn’t be calculated a priori.
Solution of integration problems using functional approach. Analog computation. An analog computational model is based on a paradigm of analogy. Here, the variables are not discrete but continuous; the solutions aren’t precise (more error prone) but at least immediate (fast). A typical analog solution: 
If we digitize this, we get the following: 
And the final digitized specification would be 
Normally, we think A numerical solution  a digital solution
Interpolated solution What if we get a direct analog solution? Example. 
here, integrator is a device which outputs A typical problem may appear more involved. Example.
Well, we would need two integrators, multiplier, several summands, …  Download 18.26 Kb. Share with your friends:
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