Project #106287 - Paraphrase

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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