computation

A. COMPUTATION
Computation can actually be interpreted as a way to find the partial derivative problem solving input data using an algorithm. This is what is called the theory of computation, a sub-field of computer science and mathematics. For thousands of years, and computational calculations are generally done using pen and paper, or chalk and slate, or done mentally, sometimes with the help of a table. But now, most computing was done using a computer.
In general, computational science is a field of science that has the attention on developing mathematical models and numerical solution techniques and the use of computers to analyze and solve the problems of science (science). In practical use, usually in the form of the application of computer simulation or various other forms of computing to solve problems in various fields of science, but in its development is also used to discover new principles are fundamental in science.
This field is different from computer science (computer science), who studied computing, computer and information processing. This field is also different from the theory and experiment as a traditional form of science and scientific work. In natural science, computational science approach can provide new insights, through the application of mathematical models in a computer program based on the theoretical basis that has been developed, to solve real problems in the sciences. Computational science is a branch of computational science. In general, computational science examines aspects of computing for applications or solve problems in other sciences, like physics, chemistry, biology and others.
Computational Physics task begins with a discussion of some numerical methods are often used to menyelesaian problems in physics, namely Newton-Raphson method, Simpson's method, Euler method, Runge-Kutta Order 4, and finite difference methods. Furthermore, these methods are used in solving problems in computational physics. In this section made the steps that need to be done to solve a given problem in order to facilitate the settlement of issues in computing. There are four steps that need to be done, the first analysis, specification issues, and program design. Second, data organization, algorithm design, and flowcharts. Third, the encoding program. Fourth, the program execution. After the fourth step is done, then performed a discussion of the results obtained. Computational Physics and science discipline that incorporates physics, numerical analysis, and computer programming, has facilitated researchers in processing experimental data and is not linear, experts said Computational Physics Prof. Dr.Drs University of North Sumatra. Muhammad Zarlis, MSc, Thursday. In his paper, Professor of Computational Physics Permanent Faculty of Mathematics and Natural Sciences University of North Sumatra, said in a simulation experiment Computational Physics, mathematical models of non-linear, and nonsimetri can be solved through the help of numerical methods in the form of computer programs. Thus, the existence of experimental physics, theoretical physics and computational physics is mutual support in research and development fields of physics, he said.According Zarlis who earned his third-Strata of the Universiti Sains Malaysia, Computational Physics is an integral part of developmental problems or physical symptoms and ability to anticipate them by using a computer device. The application of computers in the field of physical sciences are seen on solving the problems of complex analytic and numerical work to solve interactively. He further explained the computer is the result of high-tech products in recent years have often found, used, and utilized in various fields of activities in physics laboratories in both the public and private universities. Use of this computer increase even more after it manufactures various types of computers that cost relatively cheaper.
Field experience shows that the use of computers in labs is limited to typing or a particular data processing, in other words the use of computers as a versatile tool that is not maximized. When viewed from the academic staff, he said, still found many teachers are still reluctant in using computers, while the computer is as a tool for the development of computational physics.B. Partial Differential EquationsIn mathematics, partial differential equations (PDE) is a type of differential equation, which involves an unknown function of several independent variables and partial derivatives associated with these variables. Partial differential equations are used to formulate, thus helping the solution, problems involving functions of several variables, such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Different physical phenomena may have an identical mathematical formulations, thus governed by the same underlying dynamics. They found a generalization of the stochastic partial differential equations. Just as ordinary differential equations model systems are often dynamic, partial differential equations are often multidimensional model of the system. Parabolic partial differential equations.A partial differential equation (PDE) for the function u (x 1, ... xn) is in the form

 F is a linear function of u and its derivatives if, by Replacing u with v + w, F can be written as F (v) + F (w), and if, by Replacing u with me, F can be written as F is a linear function of u and its derivatives if, by replacing u with v + w, F can be written as F (v) + F (w), and if, by replacing u with me, F can be written as
If F is a linear function of u and its derivatives, then the PDE is linear. Typical examples of linear PDEs include heat equation, the wave equation and Laplace equation.A PDE is relatively simple
 This relationship indicates that the function u (x, y) is independent of x. Therefore the general solution of this equation is
 where f is an arbitrary function of y. The analog of ordinary differential equations is
 which has the solution
 where c is any constant value (independent of x). Both of these examples illustrate that the general solution of ordinary differential equations (Odes) involves an arbitrary constant, but the solution PDEs involving arbitrary functions. A solution of the PDP is generally not unique; additional conditions should generally be determined at the boundary region where the solution is defined. For example, in the simple example above, the function f (y) can be determined if u is determined on the line x = 0.
Parabolic partial differential equation is a kind of second-order partial differential equations (PDE), describing a family of sorts of problems in science including heat diffusion, ocean acoustic propagation, the physical or mathematical system with a variable time, and which behaves essentially like the spread of heat through solid objects.A partial differential equation in the form

    Au_ {xx} + {xy} + {Bu_ Cu_ yy} + \ cdots = 0is parabolic if it meets the conditions B ^ 2-.4ac = 0. This definition is analogous to the definition of parabolic planar. A simple example of a parabolic PDE is one-dimensional heat equation,

    ku_ u_t = {xx},
where u (t, x) is the temperature at time t and at position x, and k are constants. The ut the symbol denotes the partial derivative of the variable time t, and also uxx is the second partial derivative of x.
This equation states that the temperature is approximately at a certain time and the point will go up or down at a rate proportional to the difference between the temperature at that point and the average temperature near the point of it. The uxx quantity measures how far from satisfying the temperature of an average property value of harmonic functions. A generalization of the heat equationu__ t = Luwhere L is the operator of second order elliptic (implying L must be positive also, a case where L is non-positive described below). Such systems can be hidden in an equation in the form

    \ Nabla \ cdot ((x) \ nabla u (x)) + b (x) ^ T \ nabla u (x) + cu (x) = f (x)
if the matrix valued function (x) has a kernel of dimension 1.
Solution
Based on broad assumptions, parabolic PDEs such as those given above has a solution for all x, y and t> 0. Equation of the form ut = L (u) is considered a parabola if L is a function (possibly nonlinear) of u and the first and second derivatives, with some further conditions on L. Thus the nonlinear parabolic differential equations, solutions exist for a short time but can explode in a singularity in a finite amount of time. Therefore, the difficulty in determining a solution for all time, or more generally study the singularities that arise. It is in general quite difficult, as in the solution of the Poincaré conjecture through the flow Ricc  

REFERENCES

GD Smith, "Numerical solution of partial differential equations: finite difference methods", Clarendon Press (1978)
C. Johnson, "Numerical solution of partial differential equations by the finite element method" Cambridge Univ. Press (1987) Press (1987)
  AR Mitchell, DF Griffiths, "The finite difference method in partial differential equations", Wiley (1980)
  L. Lapidus, GF Pinder, "Numerical solution of partial differential equations in science and engineering", Wiley (1982)
AR Gourlay, "hopscotch, a fast second order partial differential equation solver" AW Peaceman, HH Rachford, "The numerical solution of parabolic and elliptic differential equations" SIAM J. , 3 (1955) pp. 28-41