Consider the equation APPLYING THE Taylor series for smoth function, we get: ... provided 'u' is twice differentiable and the series is applied for t > O. Making use of the a-notation, we can have: Now we can split the time interval into small parts. Let us take tm as an integer multiple of t. By invoking numerical approximation, we now consider vm as the approximation of u(tm) Let v m be a known quantity. Then v m+1 can be computed.

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So we can write: Using our defination:From this, v m+l is evaluated. This is a simple process using the Euler method. The advantage associated is that it can be implemented on a computer for any function. We shall deal with more complex problems and their solutions, using computational methods, in the forthcoming articles. Please note that you can directly apply some of these methods when you deal '0-th problems in kernel programming. But for others, you may need to 'customise' the method to suit the defined problem.