Numerical Solutions of ODEs in Engineering
EEE 244-5: Numerical solution of
Ordinary Differential Equations (ODEs)
Engineering Problem Solution
•
Start with physical laws (Faraday’s, Ohm’s,
KVL, KCL, Maxwell’s equations)
•
Generation of differential equation for the
specific problem
•
Numerical or analytical solution of differential
equation to obtain the unknown variables,
such as current, voltage, power.
3
ODE – Initial value problem
•
First order ODE with initial condition
•
Several techniques are available to solve for the
function
y(x)
4
ODE - Example
•
To solve an RL circuit, we apply KVL around the
loop and obtain a differential equation:
•
The aim is to solve for the unknown current
i(t),
given the initial condition
i(t=0)= i(0)
•
Numerical solution is practical to solve higher order
ODEs
Taylor’s Series
Euler’s method of solving first order ODE
•
To solve the ODE:
•
We use the first two terms of the Euler series
h
=
x
i+1
– x
i
,
is the step size
•
Solve for
y
using values of i=0,1,2….
7
Error in Euler’s method
8
Modified Euler’s method
•
Predictor-corrector algorithm
•
Reduces error in Euler’s method
Matlab/Python commands so solve ODEs
•
Matlab/Python have the command
ode
to
solve the first order ODE:
10
Solution of second order ODE
•
To solve the second order ODE:
•
Consider Taylor’s series (first 3 terms) for
y
i+1
and
y
i-1
; we obtain
y
as follows:
Matlab/Python commands so solve ODEs
•
Matlab/Python commands to solve the
second order ODE: (for Python define the
function f(x,y) as f(x,y[0])
This content discusses the practical application of numerical methods for solving Ordinary Differential Equations (ODEs) in engineering problems. Starting from physical laws, it covers the generation of differential equations, numerical and analytical solutions, and techniques for solving initial value problems and higher-order ODEs. Various methods such as Euler's method and Taylor series are explained, along with MATLAB/Python commands for ODE solutions. The focus is on understanding and implementing numerical techniques for solving ODEs in practical scenarios.
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Presentation Transcript
EEE 244-5: Numerical solution of Ordinary Differential Equations (ODEs)
Engineering Problem Solution Start with physical laws (Faraday s, Ohm s, KVL, KCL, Maxwell s equations) Generation of differential equation for the specific problem Numerical or analytical solution of differential equation to obtain the unknown variables, such as current, voltage, power.
ODE Initial value problem First order ODE with initial condition dy = ( , ) f x y dx with = initial condition ) 0 ( y IY Several techniques are available to solve for the function y(x) 3
ODE - Example To solve an RL circuit, we apply KVL around the loop and obtain a differential equation: di V R = i dt L L The aim is to solve for the unknown current i(t), given the initial condition i(t=0)= i(0) Numerical solution is practical to solve higher order ODEs 4
Eulers method of solving first order ODE To solve the ODE: dy = ( , ) f x y dx with = initial condition ) 0 ( y IY We use the first two terms of the Euler series h = xi+1 xi ,is the step size Solve for y using values of i=0,1,2 .
Modified Eulers method Predictor-corrector algorithm = + P i + Predictor : ( , ) y y f x y h 1 i i i ) 1 + h = + + C i + P i Corrector : ( , ) ( , y y f x y f x y + 1 1 i i i i 2 Reduces error in Euler s method 8
Matlab/Python commands so solve ODEs Matlab/Python have the command ode to solve the first order ODE: ?? ??= ?(?,?) Matlab Python for x=(x1,x2) , initial condition ? = ?0 clear f=@(x,y) f(x,y) x =[xl xu] import numpy as np import matplotlib.pyplot as plt from scipy.integrate import odeint xspan=np.linspace(xmin,xmax,points) def f(y,x): return f(x,y) [x y] = ode45( f, x,y0) plot(x,y) y=odeint(f,y0,xspan) plt.plot(xspan,y) plt.show()
Solution of second order ODE To solve the second order ODE: 2 d y = ( , ) f x y 2 dx = = with initial condition ) 0 ( y ; ) 1 ( y y y 0 1 Consider Taylor s series (first 3 terms) for yi+1 and yi-1 ; we obtain y as follows: y x f h y , ( 1 = + + 2 ) 2 y y 1 i i i 10
Matlab/Python commands so solve ODEs Matlab/Python commands to solve the second order ODE: (for Python define the function f(x,y) as f(x,y[0]) Matlab Python syms y(x) dx= y0= y1= eqn = diff(y,x,2) == f(x,y); Dy = diff(y,x) dd=(y1-y0)/dx; cond = [y(0)==y0, Dy(0)==dd]; ysol(x)= dsolve(eqn,cond) x=0:dx:xlimit; y=subs(ysol); plot(x,y) import matplotlib.pyplot as plt from scipy.integrate import odeint import numpy as np def f(y,x): return(y[1],f(x,y)) y0= y1= dx= dd=(y1-y0)/dx y00=[y0,dd] xs=np.linspace(0,xlimit,no. of points)
