what are all the algorithms involved in finding a solution with numerical methods ?(adsbygoogle = window.adsbygoogle || []).push({});

an example

say we were to solve the initial value problem:

Y′ = 2x

y(0) = 0

it's so simple, you could find a formulaic solution in your head, namely y = x2. On the other hand, say we were to use a numerical technique. (yes, i know we don't know how to do this yet, but go with me on this for a second!) the resulting numerical solution would simply be a table of values. To get a better feel for the nature of these two types of solution, let's compare them side by side, along with the graphs we would get based on what we know about each one:

Notice that the graph derived from the formulaic solution is smoothly continuous, consisting of an infinite number of points on the interval shown. On the other hand, the graph based on the numerical solution consists of just a bare eight points, since the numerical method used apparently only found the value of the solution for x-increments of size 0.2.

Using numerical solutions

so what good is the numerical solution if it leaves out so much of the real answer? Well, we can respond to that question in several ways:

The numerical solution still looks like it is capturing the general trend of the "real" solution, as we can see when we look at the side-by-side graphs. This means that if we are seeking a qualitative view of the solution, we can still get it from the numerical solution, to some extent.

The numerical solution could even be "improved" by playing "join-the-dots" with the set of points it produces. In fact this is exactly what some solver packages, such as mathematica, do do with these solutions. (mathematica produces a join-the-dots function that it calls interpolatingfunction.)

when actually using the solutions to differential equations, we often aren't so much concerned about the nature of the solution at all possible points. Think about it! Even when we are able to get formulaic solutions, a typical use we make of the formula is to substitute values of the independent variable into the formula in order to find the values of the solution at specific points. Did you hear that? Let me say it again: To find the values of the solution at specific points. This is exactly what we can still do with a numerical solution numerical methods and errors

interpolation

numerical differentiation

numerical integration

solution of algebraic and transcendental equations

numerical solution of a system of linear equations

numerical solution of ordinary differential equations

curve fitting

numerical solution of problems associated with partial differential equations fixed point iteration method

bisection and regula false methods

newton raphson method etc.

Finite differences operators

numerical interpolation

newton’s and lagrangian formulae

part i

newton’s and lagrangian formulae

part ii

interpolation by iteration

numerical differentiaton

numerical integration

solution of system of linear

equations

solution by iterations

eigen values

taylor series method

picard’s iteration method

euler methods

runge – kutta methods

predictor and corrector methods

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# I The algorithms involved in finding a solution with numerical methods

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