Comparing Eulers Method with Numerical Methods - Thesis Example

Nonetheless, differentiation of equation expressing these systems and shapes was noted to be quite complex. Moreover, these equations are only capable of describing extremely large systems and shapes, so pure mathematical analysis on them is quite impossible. The complex nature of these systems led to the usefulness of numerical approximation and computer simulations. Therefore, this paper will analyze Euler’s method in differentiating these complex mathematical equations. Notably, the numerical approximation techniques that are applied in solving the differential equation were thought of and developed long before the existence of the programmable computers.

During the Second World War, people (particularly women) used mechanical calculators (in their rooms) to solve differential equations for war purposes. However, the introduction and increase in programmable computers and computer applications have decreased the cost and increased the speed, thereby increasingly easing solving the difference equations of complex systems (Kuang and Cong, 2007). For example, laptops can easily compute a long term interjectory of over one million interacting molecules.

For about five to ten years ago, this problem seemed inaccessible to even the then fastest analog supercomputers. This essay will introduce the fundamental principle of numerical approximation and relate to geometry and curved surfaces. Thereafter, it will analyze how simple geometric problems can be handled using Euler’s method. Generally, numerical differential equations are always represented in the equation or functions, f. These functions can handle a wider range of ordinary differential equation (ODEs) and partial differential equations (PDEs).

A system of ordinary differential equations can contain any number of unknown functions. However, all these functions must be a derivative of a single independent variable, t that is the same for other functions (Kuang and Cong, 2007). On the other hand, partial differential equations often have two or more independent variables. Differential Equations There are numerous ways of solving differential equations. However, there are fundamental equations upon which all other equations are built. The first order system of differential equation takes the simplest order dy/dt = f(y, t) or y? = f(x, y).

Where dy/dy represents the change in y with time and f(y, t) is a function of variables y and t. notably, there are numerous notations for the change d/dt. The most common ones include ? and y’. This equation satisfies numerical integration that means computation from initial point y0 (the initial condition) to the other successive conditions y1, y2, y3 … since differential equations cannot be solved analytically, they take an algorithm that computes the function or equation as precise as possible, that is, yn+1 from yn.

In some cases, y may be a vector while the evolution equation may be non-linear differential equations. Problem Formulation The main theme of this essay is to use first order differential to solve differential equation under certain set conditions. Consider that the fundamental differential equation of the first order of ordinary differential equation is dy/dx = f(x, y) with the initial boundary condition being y(x0) = y0. Approximate the function y(x) over the sample values of xn