Numerical solution of Differential and Integral Equations

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Numerical solution of Differential and Integral
Equations

• PSCi702
• October 19, 2005

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Differential Equations

• Equations where the dependent variable appears as
  well as one or more of its derivatives.
• The highest derivative present determines the
  order of differential equation.
• The highest power of the dependent variable or
  its derivative sets the degree of the
  differential equation.

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Differential Equations

• Y-Y0 (1st order)
• Y(t)-Y(t)exp(t) (1st order)
• Y(t)-Y(t)t2 (1st order)
• YY-2Y0 (2nd order)
• YY-2Y0 (3rd order)
• YY20 ( 2nd degree, 1st order)
• Y(t)(Y(t)-Y(t))2 t ( 2nd degree, 2nd order)

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Differential Equations

• Higher order equations can be reduced to a system
  of first order equations.

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Differential Equations

• When solving differential equations, the final
  answer has a constant of integration in it.
• If all the constants of integrations are
  specified at the same place, they are then called
  initial values and the solution is called initial
  value problem.
• If the initial values are not given at the same
  place and are specified at different locations,
  then the solution to the problem is called
  boundary value problem.

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Solution to Differential Equations

• Start the solution at the value of the
  independent variable for which the solution is
  equal to initial values.
• Proceed step by step by changing the independent
  variable and obtaining solution across the
  required range.
• Since most methods use local polynomial
  approximation methods, stability becomes an issue.

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One Step Methods

• Picards Method

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Example

• Use Picard iteration to find the solution of

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Example
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Example

• The exact solution is

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Runge-Kutta

• The method doesnt rely on polynomial
  approximation.
• Solution can be presented by a finite taylor
  series of the form

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Runge-kutta
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Runge-Kutta
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Error Estimate

• If solution is monotonically increasing, then the
  error is increasing as well due to truncation.
• In oscillatory solutions, the truncation error
  introduces a phase shift.
• The general accuracy can not be arbitrarily
  increased by decreasing the step size. While it
  will reduce the truncation error, it will
  increase the effects of round-off error.

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Error Estimation
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Example
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Example
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Example
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Predictor-Corrector Method

• By using the solution at n points, we can fit an
  (n-1) degree polynomial.
• The predictor part extrapolates the solution over
  some finite range h based on the information at
  prior points and inherently unstable.
• The corrector part makes correction at the end of
  the interval based on some prior information.

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Predictor-Corrector Method
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Predictor-Corrector Method
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Predictor-Corrector Method
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Systems of Differential Equations
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Systems of Differential Equations
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Systems of Differential Equations

• In vector form
• Where consists of elements which are
  functions of dependent variables yi,n and xn.
• A set of basis solutions is simply a set of
  solutions, which are linearly independent.
• Consider a set of m linear first order
  differential equations where k values of the
  dependent variables are specified at x0 and (m-k)
  values corresponding to the remaining dependent
  variables are specified at xn.

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Systems of Differential Equations

• solve (m-k) initial value problems starting at x0
  and specifying (m-k) independent, sets of missing
  initial values so that the initial value problems
  are uniquely determined. Let us denote the
  missing set of initial values at x0 by

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• The columns of are just the
  individual vectors
• Matix A will have to be diagonal to always
  produce
• So one can choose
• So the missing initial values will be

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Systems of Differential Equations
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Integral Equations

• Equations can be written where the dependent
  variable appears under an integral as well as
  alone.
• Such equations are the analogue of the
  differential equations and are called integral
  equations.
• It is often possible to turn a differential
  equation into an integral equation which may make
  the problem easier to numerically solve.

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Integral Equations
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Integral Equations

• The parameter K(x,t) appearing in the integrand
  is known as the kernel of the integral equation.
• Its form is crucial in determining the nature of
  the solution. Certainly one can have homogeneous
  or inhomogeneous integral equations depending on
  whether or not F(x) is zero. Of the two classes,
  the Fredholm are generally easier to solve.

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Integral Equations