The subject extends the notion of extremum conditions of a function in one variable to such conditions of a function of functions (dependent variable and its derivatives) in the form of a definite integral. The necessary condition introduces Euler equation which results in an ordinary differential equation whose solution provides the desired extremal. The sufficient part is discussed using Legendre and Jacobi conditions. The fascinating variational principle paves the way to find the curve (line) of shortest distance between two non-intersecting curves in a plane. The book covers these topics in detail supported by figures and exercises. The book also lists some direct (approximate) methods to solve boundary value problems containing ordinary/partial differential equations by variational and residue methods, some of them being of immense importance in the treatment of finite element numerical methods. Variety of disciplines being used in the subject, are given in brief, in respective appendices.
Naveen Kumar.: Dept. of Mathematics, Faculty of Science Banaras Hindu University, Varanasi, India
Variational Problems with Fixed Boundaries: Functional/Necessary condition of extremum/ Euler equation/ Euler-Poisson equation/ Euler-Ostrogradsky equation/ Euler equation in parametric form/ Invariance of Euler equation/ Other forms of boundary conditions/ Isoperimetric problems/ Principle of reciprocity/ Exercises / Variational Problems with moving boundaries: Moving boundaries in explicit form/ Moving boundaries in implicit form/ One side variation/ Exercises / Sufficient conditions of extremum: Higher order variations/ Sufficient condition for extremum/ Jacobi equation and Jacobi conditions/ Exercises / Direct Methods: Ritz method/ Ritz method for computing eigen values/ Ritz method for boundary value problems/ Galerkin method/ Collocation method/ Least square method/ Kantorovich method/ Finite difference method / Appendix - A: Ordinary differential equations / Appendix - B: Finite difference methods / Appendix - C: Eigen value and eigen value problem / Appendix - D: Gaussian elimination method