Tutorial on Solving Boundary Value Problems (BVPs)

Introduction

• Continuation of series on differential equations.
• Focus on boundary value problems (BVPs).
• Comparison with initial value problems (IVPs).

Key Differences Between BVPs and IVPs

• IVPs: Initial conditions provided for y and y' at the beginning of the time interval.
• BVPs: Conditions are given at the boundaries of the interval.

Solving BVPs Using bvp4c Solver

• General Form: sol = bvp4c(f, bc, solinit)
  • sol: Solution object
  • f: System of equations
  • bc: Boundary conditions
  • solinit: Initial guess
• Arguments:
  1. System of equations f
     • States: y and y'
     • dsdt defined using state variables.
     • Example: y'' = -exp(y) corresponds to dsdt = [y'; -exp(y)]
  2. Boundary conditions bc
     • Defined such that right-hand side is zero.
     • Example boundaries at a (0) and b (1): bc = @(ya, yb) [ya(1); yb(1)]
  3. Initial guess solinit
     • Discretize the interval using linspace.
     • Example initialization with 5 points all set to zero: solinit = bvpinit(linspace(0, 1, 5), [0 0])
• Running the BVP solver and plotting results.
  • Result stored in sol.x and sol.y.
  • Example plotting command: plot(sol.x, sol.y)

Finding Multiple Solutions

• Use different initial guesses to find different solutions.
• Example: Multiply initial guess values to detect another solution.

Advanced BVP: Including Parameters

• Example with first-order equation and unknown parameter λ.
  1. System definition: f = @(t, Y, λ) [Y(2); 0] where Y = [y; λ]
  2. Boundary conditions: bc = @(ya, yb, λ) [ya - ...; yb - ...]
  3. Initial guess: solinit = bvpinit(linspace(a, b, 50), [initial_y; initial_λ])
• Run the solver and extract parameter: λ = sol.parameters
• Plotting results.

Conclusion

• Summary of BVP solving technique.
• Encouragement to like and subscribe.

Note: Always ensure boundary conditions equate to zero on the right-hand side.