Numerical analysis is one of part of mathematics. To deal with a physical problem one often tries to construct a mathematical model. These models in general lead to a differential equation or difference equation which cannot be solved analytically .in very few situations one can get analytic solution .therefore one has to adopt approximate methods or numerical methods .these methods are based on series expansions or they may be purely numerical leading to the estimation of the unknown at specific points in its interval of definition by simple arithmetic means .initial value and boundary value problems involving either ordinary differential equation or partial differential equations may be solved by such methods. These numerical solutions do not establish physical laws, in general, but give the clear picture of the dependence of desired variable on the various parameters of the problem.
The calculus of finite differences is the study of changes in the dependent variable y=f(x) with respect to the finite changes in the independent variable in finite difference we study many operators such as forward difference operators, backward difference operators, central difference operators, shift operators, averaging operator etc. We also study the interpolations for equal interval and unequal intervals .interpolation is also defined as the technique of estimating the function for any intermediate value of interpolations is the art of reading between the lines of a table. Interpolation can be think of a technique of achieving the most likely estimate of a certain quantity under certain specific assumptions. For equal interval there are two methods (i) Newton’s Greogry’s forward interpolation formula and (ii) Newton’s Greogry’s backward interpolation formula and for unequal intervals there are also two methods (i) Newton’s divided difference formula (ii) Lagrange’s method. Central difference interpolation formulae are used for interpolating the functional value near the middle of a given set of data.
Just like differential equations, the difference equations play a important role in dealing with the problems of economics, social and other science, especially in mathematical models corresponding to a given physical problem. Hence it is essential to study the difference equations. Numerical differentiation is the process of evaluating the derivative of a function at a point when the exact form of function is not .For this one can obtain the suitable polynomial which is the best fit for the given data by the means of suitable interpolation formula and then known but a set of values of that function is known.
The problem of numerical integration is solved by approximating the integrand by a polynomial with the help of an interpolation formula and then integrating the expansion between the desired limits.
Methods for finding the numerical solution of first order differential equations having numerical coefficient with given initial conditions to any desired degree of accuracy. The solution is obtained step by step through a series of equal intervals in the independent variable.
Solution of algebraic and transcendental equations is also possible in numerical technique by bisection method, Regula falsi method, Newton Raphson method, Iteration method etc.
Blog by- Ms. Akanksha Shukla
Asst. Prof., Department of Science