At the end of this course the student should be able to:
- Have a good understanding of the knowledge of the basic applied mathematics for engineers, within the wide area of the partial differential equations, which is adequate to his/her science.
- Know the new notions in the form of definitions and theorems that concern the basic contents of the course "Partial Differential Equations", in order to be able to apply them.
- Combine and make worthy of the knowledge that he/she acquired to other fields of the theoretical and applied mathematics, in which certain notions and principles of the present course are necessary and useful.
At the end of the course the student will have further developed the following skills and competences:
- Ability to demonstrate knowledge and understanding of essential concepts, principles and applications that are related to the partial differential equations of first and second (elliptic, parabolic and hyperbolic type) order.
- Ability to apply such knowledge to the solution of problems in other fields of the wide conception of theoretical and applied mathematics, related to the science of Chemical Engineering, or to the solution of multidisciplinary problems.
- Study skills needed for continuing profession development.
There are no prerequisite modules. It is, however, recommended that students have basic knowledge of the differential and integral calculus of one and many variables, of the vectors analysis, as well as of the linear algebra, which were taught in the corresponding modules "Single Variable Calculus and Linear Algebra" and "Multivariable Calculus and Vector Analysis". Moreover, it is a requisite basic knowledge in subjects of ordinary differential equations, which were taught to the corresponding module "Ordinary Differential Equations".
Concept of partial differential equation and its solution, well pose of problem, analytical and numerical approach, hybrid methods of confrontation. Linear partial differential equations of first order and use of characteristic curves to obtain general solution, Cauchy’s conditions or data and models of applied problems. Differential equations with partial derivatives of second order, main applications to modern technology and mathematical physics. Dirac’s functional and Heaviside’s function. Bessel’s and Legendre’s special functions, spherical harmonics, orthogonality and recurrence formulae. General introduction to basic integral transforms. Elliptic type equations and boundary value problems. Laplace’s and Helmholtz’s equations, solution with the method of separating the variables and eigenfunctions in Cartesian, polar, cylindrical and spherical coordinates with applications to several physical problems and spatial Fourier’s transform. Parabolic type equations (diffusion equation) and method of separation of variables, non homogeneous problems and dealing with the methods of asymptotic solutions and expansion to eigenfunctions. Brief introduction to hyperbolic type equations (wave equation) and method of separation of variables, principal concepts of wave propagation, finite and infinite string. Solution of problems of parabolic and hyperbolic type with initial and boundary conditions using the Fourier’s in space and the Laplace’s in time integral transforms.
Use of web pages and e–class.
- Teaching (2 hours/week): lectures using blackboard of the theory and its application to typical mathematical problems of Chemical Engineering.
- Recitation (1 hour/week): solving on the blackboard exercises concerning mainly mathematical applications of the science of Chemical Engineering.
Total Module Workload (ECTS Standards):
The instruction language is Greek. Written or / and oral examination (100% of final mark).
- D.G. Zill, "Differential Equations and Applications" (general editing & translation P. Vafeas, N. Labropoulos), Broken Hill Publishers Ltd, Nicosia, 2023 (Eudoxos / code 122074434).
- P.M. Hatzikonstantinou, "Mathematical Methods for Engineers and Scientists: Partial Differential Equations, Fourier Series & Boundary Value Problems – Complex Functions", Gotsis Publications, Patras, 2017 (Eudoxos / code 68379884).