Numerical Analysis: Introduction and Applications

What's Numerical Analysis?

It is an area of mathematics that deals with the creation, analysis and implementation of the algorithms that solve problems of the higher mathematics.

To put alternatively, the goal of Numerical Analysis is to design techniques that provide relevant solutions to mathematical problems.

Why is the need for Numerical Analysis?

Consider an EV charging station that charges thousands of electronic vehicles everyday and captures data like capacity of the vehicle's battery, compatible charging speed, etc in form of raw data. This raw data is mostly numeric. An analysis of this data can provide valuable insights on the day to day operations of the charging station. That's where Numerical Analysis comes into picture.

Consider the population of a city in a censes taken once in 10 years in given below

Estimate the population in the year 1995. $$Year: 1951 1961 1971
1981 $$ $$ Population
in Lakhs: 35 42
58 84 $$

In these types of problems, we needed Numerical Analysis Methods.

Using Numerical Approaches: Analysis Coronavirus disease (COVID-19)

The Coronavirus disease (COVID-19) is a global health care problem that international efforts have been suggested and discuss to control this disease. The Mathematical modeling with Computational simulations is an important tool that estimates key transmission parameters and predicts model dynamics of the disease. Three well-known Numerical Techniques for solving such equations, they are Euler's method, Runge-Kutta method of order two and order four.

Results based on the suggested numerical techniques and providing approximate solutions give important key answers to this global issue. Numerical results may use to estimate the number susceptible, infected, recovered and quarantined individuals in the future. The results here may also help international efforts for more preventions and improvement their intervention programs.

In order to estimate the effect of population quarantine, we divide the population into seven categories for simulation. Based on a Least-Squares procedure and officially published data, the estimation of parameters for the proposed model is given. Numerical simulations show that the proposed model can describe the transmission of COVID-19 accurately, the corresponding prediction of the trend of the disease is given.

Modern business on optimization methods to decide how to allocate resources most efficiently. For example, optimization methods are used for inventory controls, scheduling, determining the best location for manufacturing and storage facilities and investment strategies. Another important application is atmospheric modeling. In addition, to improve weather forecasts, such models are crucial for understanding the possible effecs of human activities on the Earth's climate.

Numerical Analysis is needed to solve such engineering problems that equations can't be solved by simple formulas.

In practical applications, an engineer obtain results in a numerical form. For example, from a set of tabulated data derived from an experiment, solutions of large system of algebraic equations,etc.

We understand in such away that, aim of numerical analysis is to provide efficient method for solving such problems arising in higher mathematics.

What are the Areas of Numerical Mathematics?

1. Algebraic and Transcendental Equations

The problems of solving non-linear equations of the type $$ f(x)=0 $$

2. Interpolation

Given a set of data values (x,y) of the function y=f(x), find the values of y for the given value of x within a range of data is called interpolation.

3. Curve fitting

When interpolation formulae yield unsatisfactory solutions, we try to fit the data along a curve to estimate the intermediate values. Also called data smoothing.

4. Numeric differentiation and integration

For a given dataset, to determine the numerical value of expressions like, $$ dy/dx, d^2y/dx^2,...,d^ny/dx^n $$

$$ I= \int y dx $$

5. Matrices and Linear systems

The problem of solving systems of linear algebraic equations, finding the eigenvalues and eigenvectors of matrices, problems in differential equations, fluid mechanics, etc.

6. Ordinary Differential Equations (ODE) and Partial Differential Equations (PDE)

In case of engineering problems that can be described through ODE and PDE and where the degree of precision is specified rather than objectively calculable.

7. Integral Equations

Problems of elasticity, electrostatics, etc.

In the numerical solutions of problems, we usually begin with a base dataset and compute to a level of acceptable outcome. The outcome is decided based on the preconditions set by the problem. The errors are recorded with possible causes: bad dataset, a flaw in numerical analysis method, etc.

Numerical Analysis Methods in Real Life

Numerical Analysis naturally finds application in all fields of engineering and physical sciences, but in the 21st century, it comes into life sciences, social sciences, medicine, business and even in the arts.

Here are the some Real Life examples:

In Engineering

In Crime Detection

In Scientific Computing

Finding Roots

In Heat Equation

Estimation of Ocean Currents

Curve Fitting of Tabular Data

Electromagnetics

Mollecular and Cellular Mechanisms of toxicity

Modeling Combustion flow in a Coal Power Plant

Air craft simulators, Nuclear Power Plant design, safety analysis, structural integrity, Bridge design, Real time control of refineries, Climate models, Weather Prediction, High rise building design, GPS, More parts of cars than one can shake a stick at. Industrial processes, Computational drug design, Optimization of all of the above, Economic models, Every kind of scientific research, Weapon design, Air crew scheduling, Dam design, MRI interpretation, Radiation therapy, Fusion magnet design.

Mathematical Modeling and Numerical simulation of polythermal glaciers

A mathematical model for polythermal glaciers and ice sheets is presented. The enthalpy balance equation is solved in cold and temperate ice together using an enthalpy gradient method. To obtain a relationship between enthalpy, temperature, and water content, we apply a brine pocket parameterization scheme known from sea ice modeling.

The proposed enthalpy formulation offers two advantages:

(1) the discontinuity at the cold‐temperate transition surface is avoided, and (2) no treatment of the transition as an internal free boundary is required. Fourier's law and Fick‐type diffusion are assumed for sensible heat flux in cold ice and latent heat flux in temperate ice, respectively. The method is tested on Storglaciären, northern Sweden. Numerical simulations are carried out with a commercial finite element code. A sensitivity study reveals a wide range of applicability and defines the limits of the method. Realistic temperature and moisture fields are obtained over a large range of parameters.