Lecture on Boundary Value Problems in Applied Computational Methods, Schemes and Mind Maps of Computational Methods

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Lecture 23

Boundary Value Problems

Shooting Method

AML702 Applied Computational Methods

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Difference between BVP and IVP

• Consider the equation for free falling object
• If we have to determine both velocity and position as function of time we need to solve the above equation and the following as system of equations:
• To solve this system of above two ODEs we need two initial conditions
• This forms an initial value problem (IVP)

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Boundary Value Problems

• In the figure below, in (a) for the two equations, 2 conditions are specified at t=0, i.e., at the same value of independent variable.
• This gives rise to an initial value problem
• In contrast to the above, in (b) the two conditions for a second order ODE are specified at two different values of t.
• This is a Boundary Value Problem

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BVP in Engineering

• The following example illustrates heat transfer in a thin rod along x by conduction and with surroundings by convection. The heat balance is given by

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The BV Problem

• The equation described in the previous slide actually is a mathematical model to determine the temperature along the axial direction of the rod.
• As it is a second order ODE, we need to have two conditions to obtain the solution. The most common situation is two temperatures maintained at the two ends of the rod:

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Shooting Method

• Shooting method converts a boundary value problem to an initial value problem.
• A trial-and-error approach is then implemented to develop a solution for the initial-value version that satisfies the given boundary conditions
• Let us convert the second order heat conduction equation to an IVP (2 first order odes)

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Shooting Method with Derivative Boundary

Conditions

• The boundary conditions discussed so far are known as fixed or Dirichlet boundary conditions.
• A common alternative is one where thee derivative is given
• As part of the solution we compute dependent variable and its derivative, incorporating derivative boundary conditions into the shooting method is relatively straightforward
• Based on guesses for the missing initial condition, we generate solutions to compute the given end condition.
• Eg. A gradient boundary condition for the heat conduction problem