EM 1040: Advanced Differential Equations

Basic Details:

Course Code: EM 1040
Credits: 3
Pre-requisites: None
Compulsory/Optional: EE

Aim :

To introduce analytical solving techniques for differential equations with constant coefficients and interpret the solutions.

Intended Learning Outcomes:

On successful completion of the course, the students should be able to;

Solve higher order ordinary differential equations with constant coefficients.
Analyze the solution of a second order ordinary differential equation with constant coefficients.
Apply matrix methods and Laplace transform in solving systems of ordinary differential equations with constant coefficients.
Obtain analytical solutions of first order linear partial differential equations using method of characteristics.
Classify second order linear partial differential equations and solve the wave equation, the Laplace equation and the heat equation.
Apply eigen function expansion method to solve non-homogeneous wave equation, non-homogeneous heat equation and Poisson equation.

Couse Content:

Second Order Ordinary Differential Equations

Spring mass damper equation: forced oscillations and resonance.

Laplace Transform

Definition, existence and properties; Laplace transform of standard functions, derivatives and integrals; inverse Laplace transform; solve ordinary differential equations with constant coefficients; discontinuous forcing functions; convolution.

Boundary Value Problems

Boundary value problem of a second order differential equation with constant coefficients using direct calculation; Euler Bernoulli equation and Macaulay’s Bracket method.

Systems of ODEs

converting higher-order differential equations to a system of first-order differential equations; eigenvalue eigenvector method, matrix exponential method, decoupling; phase plane analysis.

First order linear partial differential equations

Partial differential equations as a mathematical model; Method of characteristics.

Second order linear partial differential equations

classification: hyperbolic, parabolic and elliptic equations; introduction to Fourier series; method of separation of variables: wave equation, heat equation, Laplace equation on finite rectangular domains with homogeneous and non-homogeneous boundary conditions; d’Alermbert solution; Eigen function expansion method: non-homogeneous heat and wave equations

Time Allocation (Hours):

Lectures

Tutorials

Recommended Texts:

R.K. Nagle, E.W. Saff, A.D. Snider, “Fundamentals of Differential Equations”, 8th edition (2012), Pearson Education.
E. Kreyszig, “Advanced Engineering Mathematics”, 9th edition (2010), John Wiley & Sons, Inc.
Jiří Lebl, ” Differential Equations for Engineers”, Open Education Resource (OER) LibreTexts Project (https://LibreTexts.org).
Walter A. Strauss, ” Partial Differential Equations”, 2nd edition (2007), John Wiley & Sons, Inc.

Assessment:

In - course: