Numerical Analysis and its use in Communication and electronics engineering!

 

Introduction About Numerical Analysis:

 

Numerical analysis could generally be defined as the field of mathematics and computer science that creates, analyzes, and implements algorithms for solving numerically the problems of continuous mathematics. These problems arise from life applications of algebra, geometry and calculus, and they include variables that differ continuously, these problems happen all over the natural and social sciences, engineering, medicine, and business. Through the past half-century, the growth in power and obtainability of digital computers has led to an increase in the usage of genuine mathematical models in science and engineering. And since the increasing complexity, the numerical analysis has been more needed to solve these more detailed mathematical models of the world. The field of numerical analysis differs from rather theoretical mathematical studies to computer science issues.

 

Importance of numerical analysis in the field of mathematics and engineering:

 

The practical data often comes in the form of discrete data points. These are a result of a measurement you took by hand, a sensor reading, etc., but this raw data doesn’t typically contain all the information you wanted from the physical system.
Here are a couple practical examples using discrete data points and numerical methods:
·        Regression/interpolation: find a polynomial model of the coefficient of thermal expansion of a material. This changes significantly with temperature in some materials. We do this as an exercise in the undergrad class at my university.
·        Numerical differentiation: given position and time data, you can find velocity, acceleration, and/or jerk. I’ve done this as part of data analysis in my research.
 
So, we conclude that the importance of numerical analysis in mathematics and engineering is that it gives all the information need from whatever system needed from.

 

Methods of numerical analysis:

 

There’s so many uses for numerical methods; it is impossible to list them all. But essentially, we can cover first the basic math problems they can be used for, which are often:
·        Root finding and numerical optimization.
·        Estimating the solution to a set of linear and nonlinear.
·        Building models based on data, be it through interpolation, Least Square, or other methods.
·        Solving differential equations.
·        Computing integrals and derivatives.

 

Exact Solution and Approximate Solution:

 

Ordinary differential equations, partial differential equations, integral equations and other equations are encountered in various fields of mathematics, physics, mechanics, chemistry, biology, economics, and numerous applications. Exact solutions to mathematical equations play an important role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural science. Exact solutions of nonlinear differential equations graphically demonstrate and allow unraveling the mechanisms of many complex nonlinear phenomena. Furthermore, simple solutions are often used in teaching many courses as specific examples illustrating basic tenets of a theory that admit mathematical formulation. Even those special exact solutions that do not have a clear physical meaning can be used as test problems to verify the consistency and estimate errors of various numerical, asymptotic, and approximate analytical methods. Importantly, exact solutions can also serve as a basis for perfecting and testing computer algebra software packages for solving differential equations.
As for approximate solutions, they are found where there is difficulty in finding the exact solution where the problem is intractable. Exact solutions are of great educational, academic, theoretical, comprehension and validation value. But their quantitative results for a given practical problem at hand are at best approximate due to an inevitable mismatch between ideal theory and observable reality. In practice, when quantitative results for a given real-world problem are required, the numerically approximate solution may often be better (more accurate, faster, comprehensive, correct, efficient, cost-effective) than the exact solution.

Communication and Electronics Engineering:

 

Electronics and Communication Engineering is an ever-growing industry in today’s digital scenario. It is the utilization of science and math applied to practical problems in the field of Electronics & Communication. Electronic communication engineers engage in research, design, development and testing of the electronic equipment used in various communications systems. It is due to Electronics engineers that we enjoy such modern communication devices as cellular telephones, radios and television. The graduates & postgraduates are qualified for professional practice or to work in several areas of specialization. The students are prepared as professional engineers through an education in fundamental principles presented in the context of real application and design. In addition to fundamentals of science and mathematics.
Many engineering analysis and design problems are far too complex to be solved without the aid of computers. However, the use of computers in problem solving has made it increasingly necessary for users to be highly skilled in (practical) mathematical analysis. Many methods are “iterative.” Accuracy of the result depends on how many iterations are performed. It is possible that a given method might be very slow, requiring many iterations before achieving acceptable accuracy. This could involve much computer runtime. The obvious solution of using a faster computer is usually unacceptable. A better approach is to use mathematical analysis to understand why a method is slow, and so to devise methods of speeding it up.