Paraphrase the following

*Introduction*

Numerical integration is the analysis of the motion of a spacecraft leads to ordinary differential equation with time as the independent variable. It is often impractical if not impossible to solve them exactly. Therefore, to solve differentials numerically is really important. There are different methods to solve numerical integration that are used to solve initial value problems where it is difficult to obtain exact solution.

Figure 1: ODE independent variable

An ordinary differential equation or ODE is an equation containing a function of one independent variable and its derivatives as shown in Figure 1 above. The term "ordinary" is used in contrast with the term partial differential equation which may be with respect to more than one independent variable. When it says initial value problems, it means that at all condition are specified the initial value (e.g. t = 0).

MATLAB has different functions to build in for the numerical solution. For example, first order differential equation can be solved straightforward using MATLAB syntax solver. “dsolve()” is the easiest MATLAB syntax to solve ODE as shown below with an example.

Solve y(x) = x*y using MATLB:

Figure 2: First ODE “dsolve()”

Where Dy represent first derivative and C2 is a constant.

Now in order to solve high order differential equation; there is another method than “dsolve()” to be numerical solve for ODE in high order. Therefore, MATLAB provides a number of ODE solvers; will be focusing in the second one which is “ode45” which uses a four stage Runge-Kutta method to solve a give ordinary differential equation. Range-kutta method is explicit and implicit iterative used in temporal for approximation the solution of ODE. “RK4” is the classical or simplifier which is one member of Runga-Kutta methods.

Figure 3: RK methods

*Main goals & observation*

To introduce the software MATLAB for numerical computations and in particular familiarizing with basic commands through the command window and output through the graph window

Be able to solve ODE using MATLAB syntax such as solver, and Runga-Kutta method

To understand how to build in “ode45” to use 4RK method solving ODE

To know how numerical integration are easy to solve using MATLAB software

9

Conclusion

In conclusion, the situations where it is impossible

to know the function governing some

phenomenon exactly, it is still possible to derive a reas

onable estimate for the integral of the

function based on data points. The idea is to choose a m

odel function going through the data

points and integrate the model function. One definition of a

n integral as approximation of

Ru

nga-Kutta shows that the integral method of the model fu

nction converges to the integral

of the unknown function. Theoretically, numerical inte

gration is on solid ground. There are

many practical factors that influence how well numerical i

ntegration works. MATLAB is one

of the mathwork can compute numerical functions by Runga-Kut

ta method and ode45 solver,

to solve ordinary differential equation. Nonetheless, by usi

ng common sense, together with a

solid grasp of what numerical integral means using MATLAB

and how it is related to the

geometry of the function being integrated, a creative sc

ientist, mathematician or engineer can

accomplish a great deal with numerical integration

9

Conclusion

In conclusion, the situations where it is impossible

to know the function governing some

phenomenon exactly, it is still possible to derive a reas

onable estimate for the integral of the

function based on data points. The idea is to choose a m

odel function going through the data

points and integrate the model function. One definition of a

n integral as approximation of

Ru

nga-Kutta shows that the integral method of the model fu

nction converges to the integral

of the unknown function. Theoretically, numerical inte

gration is on solid ground. There are

many practical factors that influence how well numerical i

ntegration works. MATLAB is one

of the mathwork can compute numerical functions by Runga-Kut

ta method and ode45 solver,

to solve ordinary differential equation. Nonetheless, by usi

ng common sense, together with a

solid grasp of what numerical integral means using MATLAB

and how it is related to the

geometry of the function being integrated, a creative sc

ientist, mathematician or engineer can

*Conclusion*

In conclusion, the situations where it is impossible to know the function governing some phenomenon exactly, it is still possible to derive a reasonable estimate for the integral of the function based on data points. The idea is to choose a model function going through the data points and integrate the model function. One definition of an integral as approximation of Runga-Kutta shows that the integral method of the model function converges to the integral of the unknown function. Theoretically, numerical integration is on solid ground. There are many practical factors that influence how well numerical integration works. MATLAB is one of the mathwork can compute numerical functions by Runga-Kutta method and ode45 solver, to solve ordinary differential equation. Nonetheless, by using common sense, together with a solid grasp of what numerical integral means using MATLAB and how it is related to the geometry of the function being integrated, a creative scientist, mathematician or engineer can accomplish a great deal with numerical integration.

accomplish a great deal with numerical integration

Subject | English |

Due By (Pacific Time) | 01/31/2016 08:00 pm |

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