C OMPUTATION AND S IMULATION EE A SSIGNMENT T WO By: Shimiao Cheng, Femi Adeleke, Hanieh Alirezaeeabyaneh

SOME EXAMPLES OF THE USE OF NUMERICAL SOLVERS FOR REAL - LIFE ENGINEERING / PHYSICS ODE PROBLEMS Mathematical models when simulating the behavior of physical, chemical, and biological systems often include one or more ordinary differential equations (ODES) and these equations are usually solved numerically. You have already seen acceleration a=x'' (second order derivative) and v=x′ (first order derivative) in the high school. First order Ordinary Differential Equations. In engineering applications, a first order ODE usually appears in the context of an initial value problem.

y(x0) = y0, d/dx y(x) = f(x, y(x)). The function f might be a linear or non-linear function of the independent variable x and the dependent parameter y. Euler Method, The simplest approach to solving this type of equation numerically is based on finite difference approximations to the derivative. In general, yn+1 = yn + hf(xn,yn) Effectively, we are approximating the curve y(x) by a polygon, whose first side is tangent to the curve at x0.

Runge -Kutta Method, This method uses the estimate yn+1 = yn + 1/6( A1 + 2A2 + 2A3 + A4) where the auxiliary estimates are given by A1 = f(xn, yn) A2 = f(xn +h/ 2,yn +h/ 2 A1) A3 = f(xn +h/ 2, yn +h/ 2 A2) A4 = f(xn + h, yn + hA3)

P RESENT THE RESULTS OF YOUR EFFORTS TO NUMERICALLY SOLVE ( IN M ATLAB ) THE ORDINARY DIFFERENTIAL EQUATION dy/dx = x + y, with y(0) = 0 in the range (0; 10). Note that the exact solution is y(x) = ex-x-1.

A PROJECTILE IS FIRED FROM THE ORIGIN (0 ; 0) AT ANGLE µ TO THE X - DIRECTION WITH INITIAL VELOCITY V0.

F INALLY, RESPOND ( BRIEFLY ) TO M R. O M EARA ’ S COMMENTS According to what David O’meara said, we will completely agree with him about the level of education in schools that has not met the objectives of student for their third level education. Because of this, the department of education need carefully choosing the teaching material which has to to meet the needs of the society. Another improvement which can be done is to have a better quality service in each guidance consular department. They play a major role in influencing student to pick the right field in college.

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