1

Course Objectives
Students entering in Advanced Engineering Mathematics should have a firm grasp of
Calculus, linear algebra and vector calculus. They should be able to graph functions,
integration and differentiation of functions, partial derivative of functions, optimization and
evaluation of multiple integral.
The objective of Advanced Engineering Mathematics is for students to learn the basics of
Modeling and solution of differential equations. They will study standard functions with
graph, geometrical meaning of differential equations, modeling and solution of ordinary and
partial differential equations also application of Fourier series, Fourier integral and Laplace
transform.
More generally, the students will improve their ability to think critically, to analyze a real
problem and solve it using a wide array of mathematical tools. These skills will be invaluable
to them in whatever path they choose to follow, be it as a mathematics major or in pursuit of
a career in one of the other sciences.
They will also able to apply these ideas to a wide range of problems that include the
engineering equations. The students should be able to interpret the concepts of modeling
algebraically, graphically and verbally.
After the successful completion of the course, students will be able to
• expansion of functions in terms of basic trigonometric functions.
• analyze differential equations.
• solve differential equations by using tool like Laplace transform and Fourier series.
• create a modeling of engineering problems.
• solve ODEs and PDEs
The course is designed in such a way that it can be covered comprehensively in period of
semester.



2

Introduction to Some Special Functions:
Gamma function, Beta function, Bessel function, Error function and complementary Error
function, Heaviside’s function, pulse unit height and duration function, Sinusoidal Pulse
function, Rectangle function, Gate function, Dirac’s Delta function, Signum function, Saw
tooth wave function, Triangular wave function, Half wave rectified sinusoidal function, Full
rectified sine wave, Square wave function.



3

Fourier Series and Fourier integral:
Periodic function, Trigonometric series, Fourier series, Functions of any period, Even and
odd functions, Halfrange Expansion, Forced oscillations, Fourier integral.



4

Ordinary Differential Equations and Applications:
First order differential equations: basic concepts, Geometric meaning of y’ = f(x,y)
Direction fields, Exact differential equations, Integrating factor, Linear differential
equations, Bernoulli equations, Modeling , Orthogonal trajectories of curves.
Linear differential equations of second and higher order: Homogeneous linear differential
equations of second order, Modeling: Free Oscillations, Euler Cauchy Equations,
Wronskian, Non homogeneous equations, Solution by undetermined coefficients, Solution
by variation of parameters, Modeling: free Oscillations resonance and Electric circuits,
Higher order linear differential equations, Higher order homogeneous with constant
coefficient, Higher order non homogeneous equations. Solution by [1/f(D)] r(x) method for
finding particular integral.



5

Series Solution of Differential Equations:
Power series method, Theory of power series methods, Frobenius method.



6

Laplace Transforms and Applications:
Definition of the Laplace transform, Inverse Laplace transform, Linearity, Shifting theorem,
Transforms of derivatives and integrals Differential equations, Unit step function Second
shifting theorem, Dirac’s delta function, Differentiation and integration of transforms,
Convolution and integral equations, Partial fraction differential equations, Systems of
differential equations



7

Partial Differential Equations and Applications:
Formation PDEs, Solution of Partial Differential equations f(x,y,z,p,q) = 0, Nonlinear PDEs
first order, Some standard forms of nonlinear PDE, Linear PDEs with constant coefficients,
Equations reducible to Homogeneous linear form, Classification of second order linear
PDEs.
Separation of variables use of Fourier series, D’Alembert’s solution of the wave equation,
Heat equation: Solution by Fourier series and Fourier integral


