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##### Module Notes

Faculty Member (Members):

Postgraduate, Fall Semester

*Module Type:*Specialization Courses

*Teaching Language:*English/Greek

*Course Code:*

*Module Availability on Erasmus Students:*No

##### Module Details

*Structure of Vector Spaces*: Subspaces, generating systems, span, linear independence, bases, dimension, codimension.*Linear Operators*: Matrix representation, similarity, change of bases. Kernel and range spaces, rank, nullity and invertibility. LU deocomposition.*Spectral Theory*: Characteristic quantities, algebraic and geometric multiplicity of eigenvalues, direct sum decomposition of vector spaces and of linear operators. Generalized eigenvectors and eigenspaces. Diagonal representations and Jordan canonical forms.*Introduction to Perturbation Theory*: Regular and singular perturbation theory, inner and outer solutions. Examples from chemical engineering problems.*Partial Differential Equations*: The methods of characteristics. Examples from chemical engineering problems.*Green’s Functions*: The fundamental solutions of partial differential equations.*Potential Theory*: Modeling of time invariant physical problems in isotropic and anisotropic spaces. Solution of inviscid flow and steady state problems.*Stokes Flow*: The mathematical peculiarities of elliptic equations of the fourth order.*Diffusion Theory*: Mathematical characteristics of non–invertible physical phenomena governed by parabolic equations. Fundamental solution and infinite speed of propagation. Solutions of transient diffusion problems.*Wave Theory*: Modeling the physical behavior of initial and boundary value problems governed by hyperbolic equations. Main characteristics of vibration, radiation, wave propagation and scattering problems.

#### Comments

Prerequisite for this course is knowledge of Advanced Calculus, Matrix Theory, Linear Algebraic Systems and Ordinary Differential Equations. The course will be adapted to the engineering point of view. That is, it will emphasize the mathematical interpretation and the corresponding formulation of the relative physical phenomena, avoiding any rigorous mathematical analysis. Numerous examples and applications will be the core of the course.

- G. Strang,
*Introduction to Applied Mathematics*, - N.R. Amundson,
*Mathematical Methods in Chemical Engineering: Matrices and their Applications*, Prentice Hall, 1966. - V.G. Jenson and G.V. Jeffreys,
*Mathematical Methods in Chemical Engineering*, Academic Press, 1977. - J. Crank,
*The Mathematics of Diffusion*, Oxford University Press, 1956. - H. Margenau and G.M. Murfphy,
*The Mathematics of Physics and Chemistry*, Van Nostrand, 1943. - H.–K. Ghee, R. Aris and N.R. Amundson,
*First Order Partial Differential Equations*, Volume I, Prentice Hall, 1986. - D. Ramkrishna and N.R. Amundson,
*Linear Operator Methods in Chemical Engineering with Applications to Transport and Chemical Reaction Systems,*Prentice Hall, 1985. - E. Kreyszig,
*Advanced Engineering Mathematics*, Wiley, 1983. - J.P. Keener,
*Principles of Applied Mathematics*, Addison–Wesley, 1988. - J.D. Logan,
*Applied Mathematics*, - C.M. Bender and S.A. Orszag,
*Advanced Mathematical Methods for Scientists and Engineers*, McGraw–Hill, 1984. - H. Sagan,
*Boundary and Eigenvalue Problems in Mathematical Physics*, Dover, 2012. - I. Stakgold,
*Green’s Functions and Boundary Value Problems*, Wiley, 1998. - R.E. Collins,
*Mathematical Methods for Physicists and Engineers,*Reinhold, 1968. - G.F.D. Duff and D. Naylor,
*Differential Equations of Applied Mathematics*, - H.S. Mickley, T.K. Sherwood and C.E. Reed,
*Applied Mathematics in Chemical Engineering*, McGraw–Hill, 1957.