Supplemental Unit S6. Solving Boundary Value Differential Equations

Kinetics and reaction engineering involve solving several different kinds of mathematics problems. In very simple cases these problems can be solved analytically, that is, using paper, pencil and hand-held calculator. In many cases, however, even very simple problems cannot be solved analytically. Instead they must be solved numerically using a computer. This supplement to “A First Course on Kinetics and Reaction Engineering” presents a very brief overview of numerical methods that are commonly used to solve kinetics and reaction engineering problems. The intent is not to provide comprehensive coverage of the methods; for that you should consult a book on numerical methods or take a course on the topic.

This supplemental unit describes how to numerically solve a set of coupled, first order boundary value ordinary differential equations (ODEs) of the following form, where the f's are functions of an independent variable, x, and the dependent variables, y₁ through y_n:

dy₁/dx = f₁(x, y₁, ..., y_n); a ≤ x ≤ b
dy₂/dt = f₂(x, y₁, ..., y_n); a ≤ x ≤ b
     .
     .
     .
dy_n/dt = f_n(x, y₁, ..., y_n); a ≤ x ≤ b

It also shows how second order boundary value ODEs can be re-written as sets of first order boundary value ODEs with the proper form. The values of some of the dependent variables must be known at x = a, and the values of the other dependent variables must be known at x = b. A general form for the boundary conditions is as follows:

g₁(y₁(a),...,y_n(a),y₁(b),...,y_n(b)) = 0
g₂(y₁(a),...,y_n(a),y₁(b),...,y_n(b)) = 0
     .
     .
     .
g_n(y₁(a),...,y_n(a),y₁(b),...,y_n(b)) = 0

Learning Resources

• Archive (.zip) - Contains all learning resources listed below for this unit
• Documents to Read:
  • Unit S6 Informational Reading (.pdf)
  • Example S6.1 (.pdf)
  • Example S6.2 (.pdf)
• Videos (.mov files) to Watch:
  • Unit S6 Informational Video (.mov)
  • Example S6.1 (.mov)
  • Example S6.1 (.mov)
• Other Files:
  • MATLAB Template File SolvBVDif.m and the SolvBVDif_How_To (.pdf) that gives step-by-step instructions for using SolvBVDif.m
  • MATLAB Template File SolvBVDifS.m and the SolvBVDifS_How_To (.pdf) that gives step-by-step instructions for using SolvBVDifS.m
  • S6_Example_1.m - modified version of SolvBVDif.m that was used in the solution of Example S6.1
  • S6_Example_2.m - modified version of SolvBVDifS.m that was used in the solution of Example S6.2

Teaching Resources

• In-Class Review Slides
  • Keynote 09
  • PowerPoint