The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. In mathematics, Integral equations are equations in which an unknown function appears under an integral sign. Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogenous and inhomogeneous; Fredholm and Volterra; first kind, second kind, and third kind; and singular and regular integral equations.
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering, physical sciences, social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include ordinary differential equations as found in celestial mechanics, numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.
The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.
In mathematics, an ordinary differential equation (ODE) is a differential equation dependent on only a single independent variable. The term "ordinary" is used in contrast with partial differential equations which may be with respect to more than one independent variable. The principal objectives of this course are
to understand the method of successive approximations, Lipschitz condition, and Convergence of successive approximations,
to learn the Series Solution of Second Order Linear Equations,
to know the solution techniques for solving the Systems of Differential Equations.
