A DIALOGUE ON NUMERICAL TECHNIQUES

Student: What is numerical technique?
Dr Airil: Numerical technique is a method for solving DE in an approximate but converging manner.

Student: What do you mean by approximate and converging?
Dr Airil: Approximate as opposed to analytical whilst converging means the increase in the degree of accuracy due to the increase in the number of terms or orders of the 'trial' function.

Student: What is a 'trial' function.
Dr Airil: A 'trial' function is a function that we 'guessed' to be the solution of the DE.

Student: Why we must 'guess' a function for a solution of the DE?
Dr Airil: We must guess because if we know the solution already, then we would not have the problem in the first place, would we?

Student: How numerical technique works, in other words, what is the basic idea of numerical technique?
Dr Airil: The basic idea of any numerical techniques is to convert the continuous nature of the DE into a set of algebraic simultaneous equations, then usually treated and solved by matrix operation.

Student: How this conversion is done?
Dr Airil: This is done by first, assuming for a 'trial' function which consists of coefficient/constant and the independent variables i.e. x, y, z. For example, a1+a2x+a3x^2 is a form of 'trial' function usually used. Secondly, insert this 'trial' function into the DE by conducting the appropriate differentiation. Such an insertion will turn the DE from its differential forms into ordinary function. Let's call this function as R. Thirdly, evaluate this function, R by either integrating it or by collocating it and set it to zero. The latter refers to evaluating R at certain locations say L1, L2. These evaluation process will convert the independent variables i.e. x,y,z into numerical values, leaving the coefficient/constant i.e. a1, a2, b1,b2 etc as the unknown variables which differ from previous x,y,z in the sense that these variables are constant in nature. Due to this constant property, the final product of the whole process is algebraic thus allows for its treatment in matrix forms. This is the basic idea of any numerical techniques.

Student: What are the examples of numerical techniques?
Dr Airil: As I have told you before, examples of numerical techniques are the Finite Difference, Boundary Elements and of course Finite Element. And recently, a new family of techniques has emerged called Meshfree techniques. But, as far as I have experienced it, among these techniques, Finite Element is the most robust and versatile and will survive the test of time.

Student: You keep mentioning about the matrix treatments, why?
Dr Airil: I keep mentioning about the matrix treatment because this is the essence of numerical technique. The matrix treatments or operations are beneficial in the following ways, 1) suits computer programming, 2) allows compact mathematical representation of a complex problem.

Student: Can you elaborate on the two points?
Dr Airil: On point 1), you see, matrix operation and computer programming complement each other. Whilst the knowledge of matrix has been known as early as the beginning of 1900's but its practical applications were 'retarded' I would say due to the ongoing development of the computer technology. The later establishment of computer would then 'burst' the application of matrix operation. But in turn, the development of the computer itself was very much influenced or 'guided' by the potential borne by the knowledge of matrix. If I am not mistaken, Heisenberg once said something like, matrix is the most beautiful mathematical tool and this was before the start of any works on computer development. The father of computer, John Von Neumann was a mathematician heavily influenced by matrix. So matrix and computer is like chicken and egg, you know what I mean.
On point 2), although the basic principle is not difficult, but the use of numerical technique will lead us to deal with huge and complex physical problems. But, with the availability of matrix representation, the 'length' of the equations can still be made as if it is still 'short' hence the terms compact. As a result of such a compactness, we can still 'do the calculation in our head' regardless of how big and complex the physical problem would be.

Student: Thanks.
Dr Airil: My pleasure.