Course code Mate5017

Credit points 4

Total Hours in Course160

Number of hours for lectures32

Number of hours for seminars and practical classes32

Independent study hours96

Date of course confirmation20.03.2019

Responsible UnitDepartment of Mathematics

Matemātikas katedra
## Svetlana Atslēga

Dr. math.

Fizikas katedra
## Zanda Gavare

Dr. phys.

In this course the ordinary and partial differential equations are studied at advanced level. The course provides an understanding of the modern concepts and problems of mathematical modelling. It encourages mathematical thinking, and provides insight into various mathematical applications in information technologies and other fields. This course provides students with knowledge of construction of mathematical physics equations for description of the functioning of biological and other systems. Students will gain knowledge about formal solution schemes of equations and their realization on computer. In addition, students will acquire skills in working with appropriate software such as Matlab.

The course is interdisciplinary, it combines the fields of mathematics and physics – using physiscs and mathematics knowledge students learn to construct equations that describe real processes, as well as students learn methods to solve these equations.

After successful completion of this course the student:

1. knows and is capable of proving knowledge and critical understanding of ordinary and partial differential equations, their applicability for describing the real-world processes considered in their specialty. – tests

2. is able to show understanding of relevant concepts and regularities, to perform necessary actions and operations. The student is able to use the appropriate software to make calculations. While working in a group or doing work independently, he/she is able to apply mathematical calculations corresponding to the problem of the specialty, to perform the intermediate results of the calculations and professional evaluation and interpretation of the final results, to apply knowledge in calculations for their branch of research, summarize and analytically describe the results, to evaluate the results of calculations in solving of mathematical physics problems. - independent works

1. Review of software application necessary features according to the content of this course. (8 h)

2. Mathematical modelling. Classification of mathematical models. Differential equations. Models based on ordinary differential equations. (4 h)

3. Ordinary differential equations and their types. Composing of differential equations. Solving of ordinary differential equations using appropriate software programs. Graphical representation of the solution. Solvers for ordinary differential equations (6 h)

4. Differential equation systems. Critical points. Analysis and construction of the phase plane. Examples of the application of differential equation systems in various fields of human activity (6 h)

5. 1st test: Ordinary differential equations (2 h)

6. Methods of Mathematical Physics. Mathematical field theory. Differential operators. Most common types of equations in Mathematical Physics. (6 h)

7. Examples of composing the equations. Methods of Mathematical physics problem solving. Numerical methods of Mathematical physics problem solving. (8 h)

8. The solution of parabolic type partial differential equations using finite differences. (6 h)

9. Finite difference schemes for solution of elliptic and hyperbolic (in one dimension) type equations. (6 h)

10. Finite difference scheme for solution of hyperbolic type equation in two dimensions. (4 h)

11. Experimental data processing. Method of least squares. Method of least squares for nonlinear functions. (6 h)

12. 2nd test: Partial differential equations (2 h)

The requirement is – passing the test.

The following independent works must be completed and credited, both in writing and using the appropriate application software:

1st Independent Work: Ordinary differential equations and their solvers

2nd Independent Work: Critical points. Phase plane analysis and construction

3rd Independent Work: Partial differential equations

Individual tasks must be completed and the software programs must be created for their computerized solution:

Individual Work No 1: Specific mathematical equation (parabolic, elliptical or hyperbolic)

Individual Work No 2: Application of the least squares method for approximation of given data.

The test is accumulative, without additional knowledge testing, summarizing the results of the semester:

1. all tasks of independent works are accepted by the course lecturer (at least 80% of all tasks are correctly solved)

2. each test during semester is scored at least 4 points.

3. two individual tasks are fulfilled (according to the themes of the individual tasks, two programs have been created, which student can demonstrate and explain), each task is evaluated by 0-10 points. At least 50% of the maximum score must be collected.

The student may rewrite unsuccessfully written tests during the study process at the times indicated by the course lecturer.

Independent or individual works that are not accepted at a given time can be defended at the times indicated by the course lecturer.

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http://vig.pearsoned.co.uk/catalog/academic/product/0,1144,0135113148-TOC,00.html

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Grāmatai ir jaunāka versija: Rabinovich, Semyon G. Evaluating Measurement Accuracy: A Practical Approach. Springer, 2017. [skatīts 17.04.2019.] Pieejams SpringerLink: https://www.springer.com/gp/book/9783319601243

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1. Said Elnashaie, Frank Uhlig. Numerical techniques for Chemical and biological engineers using MATLAB. A simple bifurcation approach. Springer, 2007. – 590 p.

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Course for master study programme “Information Technologies”.