MATH566 Lesson 25: Numerical ODE - BVP

Overview

• Boundary value problems

• Linear boundary value problems

• Nonlinear boundary value problems

Boundary value problems

• It is known that an ODE of order $n$ requires $n$ boundary conditions

• If all boundary conditions are specified at a single point an IVP is obtained, e.g.,

  ${\displaystyle y^{\text{'}\text{'}}+y^{\text{'}}+y=t,y\left(0\right)=y_0,y^{\text{'}}\left(0\right)=y_0^{\text{'}}}$

• For order $n>1$ boundary conditions can be given at different points, a BVP

  ${\displaystyle y^{\text{'}\text{'}}+y^{\text{'}}+y=t,y\left(0\right)=u,y\left(1\right)=v}$

• Boundary value problems give rise to mathematics of wide applicability, in particular Sturm-Liouville problems that define the special functions of mathematics (MATH528, MATH529)

• Sturm-Liouville problem for $y:[a,b]\to ℝ$, $r,q,p,r^{\text{'}}\in 𝒞[a,b]$, $\lambda \in ℝ,$

  ${\displaystyle \begin{array}{c}\frac{d}{dx}\left[r\left(x\right)y^{\text{'}}\right]+[q\left(x\right)+\lambda p\left(x\right)]y=0,\\ A_{1}y\left(a\right)+B_1{y}^{\text{'}}\left(a\right)=0,A_{2}y\left(b\right)+B_2{y}^{\text{'}}\left(b\right)=0,\\ A_1{B}_1\ne 0,A_2{B}_2\ne 0.\end{array}}$

Linear boundary value problems

• Consider a linear differential operator $ℒ$ acting on $y$, $ℒy=f$

  ${\displaystyle ℒ(\alpha u+\beta v)=\alpha (ℒu)+\beta (ℒv)}$

• Examples

  • $ℒ=\frac{d^2}{d{t}^2}+1$,$f\left(t\right)=t$, $y^{\text{'}\text{'}}+y=t$, $y\left(0\right)=0$, $y\left(1\right)=1$, $t\in [0,1]$

  • $ℒ=\frac{d^2}{d{t}^2}+p\left(t\right)\frac{d}{dt}+q\left(t\right)$, $f\left(t\right)=0$, $y^{\text{'}\text{'}}+p\left(t\right)y^{\text{'}}+q\left(t\right)y=0$

• Numerical solution of linear boundary valeu problems is typically carried out by approximation of the derivatives to obtain a linear system, e.g. for $y^{\text{'}\text{'}}+y=t$

  ${\displaystyle h=1/m,x_i=ih,i=0,1,\dots ,m}$ ${\displaystyle \frac{y_{i+1}-2y_i+y_{i-1}}{h^2}+y_i=x_i,i=1,2,\dots ,m-1}$

• For $y^{\text{'}\text{'}}+p\left(t\right)y^{\text{'}}+q\left(t\right)y=0$ with $y\left(0\right)=0$, $y\left(1\right)=1$, $t\in [0,1]$ obtain

  ${\displaystyle \frac{y_{i+1}-2y_i+y_{i-1}}{h^2}+p\left(x_i\right)\frac{y_{i+1}-y_{i-1}}{2h}+q\left(x_i\right)y_i=x_i,i=1,2,\dots ,m-1}$