MAT115 Differential and Difference Equations L T P C

3 0 2 4 Version No. : 1.00 Course Prerequisites

: MAT101/MAT114 Multivariable Calculus and Differential Equations

Objectives : This course is designed to give a comprehensive coverage at an introductory level to the subject of ordinary differential equations and difference equations. Matrix methods and eigenvalue problems are integrated in to the course. Sufficient emphasis is laid on mathematical modeling and analysis of simple engineering problems. Expected Outcome: At the end of this course, the students are expected to know how to model simple physical problems in the form of a differential and difference equations, analyze and interpret the solutions. Further the students are expected to acquire necessary background in matrix methods and Eigen value problems so as to appreciate their importance to engineering systems.

Unit 1 Matrix methods to Linear Differential Equations 9L+4P hours The eigen value problem- eigen values and eigen vectors - properties of eigen values and eigen vectors-Cayley-Hamilton theorem and its applications- symmetric matrices -similarity of matrices - diagonalisation of a real symmetric matrix-quadratic form.Solution of equations of type X11 + AX=0 - reduction of nth order equation to a system of first order equations by diagonalization.

Unit 2 Power Series Solutions 9L+4P hours The Strum-Liouville Problem-orthogonality of eigen functions- Bessel’s and Legendre’s equations- power series solutions – method of Frobenius.

Unit 3 Fourier Series 9L+4P hours Fourier series -Euler’s formulae- Dirichlet’s conditions - change of interval- half range series – RMS value – Parseval’s identity – computation of harmonics.

Unit 4 Difference Equations and Z-transforms 9L+ 4P hours Difference equation-first and second order difference equations with constant coefficients-Fibonacci sequence-solution of difference equations-complementary functions - particular integrals by the method of undetermined coefficients. Z-transform-relation between Z-transform and Laplace transforms - Z-transforms of standard functions-inverse Z-transforms : by partial fraction method, by convolution method- solution of simple difference equations using Z-transforms.

Unit 5 Applications of Differential Equations 9L+4P hours First order equations: Newton’s law of cooling – radioactive decay, L-R and C-R circuits-Equation of motion for a particle in gravitational field – Terminal velocity. Second order equations: Free un-damped and damped vibrations, Forced oscillations-Resonance phenomenon, series LCR circuit - Model of a vibrating systems with two masses – Solutions by matrix methods.

Text Books: 1. Erwin Kreyszig, Advanced Engineering Mathematics, 10th Edition, Wiley India Pvt. Ltd. (2013). Reference Books

1. B. S.Grewal, Higher Engineering Mathematics, 42nd Edition. Khanna Publications,(2013).

2. Michale D. Greenberg, Advanced Engineering Mathematics, 2nd Edition, Pearson Education, First Indian reprint (2002).

3. Peter V. O’ Neil, Advanced Engineering Mathematics, 5th Edition, Thomson, Book/Cole. (2007).

Proceedings of the 34th Academic Council [23.12.14] 82

MAT115 Differential and Difference Equations

(Lab experiments)

Exp. No.

Title

Aim

1.

A brief review of MATLAB, Eigen values and Eigen vectors, properties

To find the eigen values and eigen vectors for a given matrix using MATLAB

2.

Cayley-Hamilton theorem and Diagonalization of Matrix

To find the diagonalized matrix for a given matrix using MATLAB

3.

System of second order Differential Equations of the form X’’+AX=0

To write MATLAB code for system of second order differential equations of the form X’’ +AX=0 using diagonalization.

4.

Series solution for ODE To find the power series solution and visualize it for the second order initial value problem with ordinary point using MATLAB.

5. Fourier series To write the MATLAB code for finding Fourier coefficients.

6. Harmonic Analysis To write MATLAB codes to find the few harmonics of Fourier series and visualize the series when f(x) is given as data points.

7. Solving second order Difference Equations

To write the MATLAB code to find the solution for the second order linear difference equation with constant co-efficient.

8.

Solving of first and second order difference equations using z-transform

To solve and visualize Difference equation using Z – Transform in MATLAB.

9. Applications of Second order Differential Equations

To know how the second order differential equations can be applied to solve problems concerning the vibrations of springs using MATLAB.

10. Applications of Second order Differential Equations.

To know how the second order differential equations can be applied to solve problems concerning the analysis of electric circuits using MATLAB.

Proceedings of the 34th Academic Council [23.12.14] 83