Complex Variables and Partial Differential Equation (MAT3003)

Credits : 4

NOTE:

The aim of this course is to present a comprehensive, compact and integrated treatment of two most important branches of applied mathematics for engineers and scientists namely the functions of complex variable and Partial differential equations in finite and infinite domains.

MODULE 1

Topics Included: Analytic Functions: Complex variable, Analytic functions and Cauchy, Riemann equations, Laplace equation and Harmonic functions Construction of Harmonic conjugate and analytic functions Applications of analytic functions to fluid-flow and Field problems. Videos to Watch: According to the playlist 1.1 to 1.3

MODULE 2

Topics Included: Conformal and Bilinear transformations: Conformal mapping, Elementary transformations-translation, magnification, rotation, inversion. Exponential and Square transformations, Bilinear transformation, Cross-ratio-Images of the regions bounded by straight lines under the above transformations. Videos to Watch: According to the playlist 2.1 to 2.3

MODULE 3

Topics Included: Power series: Functions given by Power Series, Taylor and Laurent series – singularities – poles – Residues. Videos to Watch: According to the playlist 3.1 to 3.2

MODULE 4

Topics Included: Complex Integration: Integration of a complex function along a contour , Cauchy-Goursat theorem, Cauchy’s integral formula, Cauchy’s residue theorem, Evaluation of real integrals, Indented contour integral. Videos to Watch: According to the playlist 5.1

MODULE 5

Topics Included: Partial Differential equations of first order: Formation and solution of partial differential equation - General, Particular, Complete and Singular integrals - Partial Differential equations of first order of the forms: F(p,q)=0, F(z,p,q)=0, F(x,p)=G(y,q) and Clairaut’s form - Lagrange’s equation: Pp+Qq = R. Videos to Watch: According to the playlist 6.1

MODULE 6

Topics Included: Applications of Partial Differential equations: Linear partial differential equations of higher order with constant coefficients. Solution of a partial differential equation by separation of variables, Boundary Value Problems, one dimensional wave and heat equations, Fourier series solution. Videos to Watch: According to the playlist 7.1

MODULE 7

Topics Included: Fourier transforms : Complex Fourier transform and properties, Relation between Fourier and Laplace transforms, Fourier sine and cosine transforms – Convolution Theorem and Parseval’s identity. Videos to Watch: According to the playlist 7.1

MODULE 8

Topics Included: Contemporary Issues: Industry Expert Lecture