Assignments for MAG6134 Uploaded.
Course Objectives: This course is intended to prepare the student with mathematical tools and techniques that are required in advanced courses offered in the applied mathematics and engineering programs. The objective of this course is to enable students to apply transforms and variation problem technique for solving differential equations and extremum problems.
Course Credit: 3.5
Laplace Transform: Review of Laplace transform, Applications of Laplace transform in initial and boundary value problems, Heat equation, Wave equation, Laplace equation.
Fourier Series and Transforms: Definition, Properties, Fourier integral theorem, Convolution theorem and Inversion theorem, Discrete Fourier transforms (DFT), Relationship of FT and fast Fourier transforms (FFT), Linearity, Symmetry, Time and frequency shifting, Convolution and correlation of DFT. Applications of FT to heat conduction, Vibrations and potential problems, Z-transform.
Hankel Transform: Hankel transforms, Inversion formula for the Hankel transform, Infinite Hankel transform, Hankel transform of the derivative of a function, Parseval's theorem, the finite Hankel transforms, Application of Hankel transform in boundary value problems.
Integral Equations: Linear integral equations of the first and second kind of Fredholm and Volterra type, Conversion of linear ordinary differential equations into integral equations, Solutions by successive substitution and successive approximation, Neumann series and resolvent kernel methods.
Calculus of Variations: The extrema of functionals, the variation of a functional and its properties, Euler equations in one and several independent variables, Field of extremals, Sufficient conditions for the extremum of a functional conditional extremum, Moving boundary value problems, Initial value problems, Ritz method.
Course learning outcomes: Upon completion of this course, the student will be able to:
1) Laplace Transformation to solve initial and boundary value problems.;
2) To learn Fourier transformation and Z transformation and their applications to relevant problems.;
3) To understand Hankel's Transformation to solve boundary value problem.;
4) Find solutions of linear integral equations of first and second type (Volterra and Fredholm)
5) Understand theory of calculus of variations to solve variational problems.
Recommended Books:
1) Simmons G.F., Differential Equations with Applications and Historical Notes, Tata McGraw Hill, (1991)
2) Gelfand I.M. and Fomin S.V., Calculus of Variations, Prentice Hall (1963).
3) Kenwal Ram P., Linear Integral Equations: Theory and Techniques, Academic Press (1971).
4) Sneddon I.N., The Use of Integral Transforms, Tata McGraw Hill (1985).
5) Churchill, R.V., Operational Mathematics, McGraw-Hill (1971)
Evaluation Scheme:
Event Weightage
Mid-Semester Test 30%
Sessionals 25%
End-Semester Test 45%