Advanced Mathematics For Engineers - Essay Example

Maclaurin series is known as a special case of Taylor series expansion at x = 0. Through Maclaurin series, a combination of functions, say those which are exponential and trigonometric in nature, may be brought altogether to acquire algebraic representations. Leibnitz’ theorem – Leibnitz’s theorem is normally applied whenever numerical methods merely work for determining solutions of first order differential equations (DEs). In particular, by Leibnitz’s theorem, second order DEs may be solved through a process of successive differentiations wherein the nth differential coefficient of standard function can be obtained by performing a series of tasks with the product rule to arrive at the intended solution for Yn.

Bessel’s and Legendre equations. Out of the studies made for the disturbances in planetary motion by Friedrich Wilhelm Bessel emerged what came to be acknowledged in the early 19th century as the first systematic analysis of solutions to the equation given by: Such an equation is called a Bessel’s equation which varies in order depending on the real constant ‘v’. . Moreover, this method had been of ample significance in the quantum mechanical model of the H-atom and is typically employed in areas of physics or engineering that tackle steady-state temperature within solid spherical objects involving the use of Laplace’s equation.

Euler, and Runge – Kutta numerical differential equation methods. Both of the principles of applying Euler method and Runge – Kutta method are vital in solving DEs of the first order. With Euler method, on one hand, restrictions are set given initial values x0 and y0, and the range of ‘x’ within which the desired solution for ‘y’ is achieved upon a number of successive iterations that follow a simple form: f (a + h) = f (a) + h [ f ‘(a) ] Iterative use of this equation proceeds until one arrives at the intended value for ‘y’ that is accurate to the extent of decimal places specified.

Similarly, the Runge-Kutta method is used for the same purpose of approximating the ‘y’ to converge to a certain value, only this time, a couple of evaluation steps are required towards a higher degree of accuracy for the results. It is necessary herein to evaluate k-values (k1, k2, k3, and k4) which must be substituted into The numbers identifying each ‘k’, as well as the YP1 and the YC1 are tabulated for a specific range of Xn. (2) Consider for the range x = 1 to x = 1.5 in increments of .

1, given the initial conditions that when x = 1, y = 2 Apply Euler Method to solve and graph the above problem Apply Euler –Cauchy Method to solve and graph the above problem Apply Runge - Kutta Method to solve and graph the above problem By Euler Method f(a + h) = f(a) + f(h) ---? y = y0 + h(y0’),