Ədədi üsullar-2Syllabus

Subject code

i İF-40170y

Credits

3

Student workload

90

Semester

Autumn semester

Academic year

2025-2026

Subject type

Mandatory

Form of education

Full-time

Language of instruction

Azerbaijani

In-class hours

a) 30 saat - mühazirə b) 15 saat - seminar

Out-of-class hours

a) 27 saat - sərbəst iş b) 18 saat - hazırlıq

“ədədi üsullar–2” fənni mürəkkəb riyazi və tətbiqi məsələlərin həlli üçün müasir ədədi metodların öyrənilməsini əhatə edir. fənn çərçivəsində runge–kutta üsulları, sonlu fərq, kollokasiya və qalerkin metodları, elliptik, parabolik və hiperbolik tip diferensial tənliklərin ədədi həlli, inteqral tənliklərin həlli üsulları, həmçinin bu metodların kompüter texnologiyalarında reallaşdırılması öyrədilir. kursun sonunda tələbələr müxtəlif ədədi alqoritmləri qurmaq və tətbiq etmək, hesablama nəticələrinin dəqiqliyini qiymətləndirmək, elmi və mühəndislik problemlərinin həllində müasir ədədi üsullardan səmərəli istifadə etmək bacarığı əldə edirlər.

• lecture
• interactive discussion
• seminar
• presentations

1. 1

   CLO 1: Justifies management measures aimed at the protection and efficient use of water resources.

2. 2

   CLO 2: Rus dilində işgüzar və akademik ünsiyyətin əsas prinsiplərini tətbiq edə biləcək. peşə və akademik fəaliyyətlə bağlı mətnləri oxuyaraq əsas və xüsusi məlumatları müəyyən edə biləcək. rus dilində rəsmi və akademik üslubda yazılı mətnlər hazırlaya biləcək. i̇şgüzar görüşlərdə, müzakirələrdə və təqdimatlarda rus dilində səmərəli kommunikasiya qura biləcək. peşəkar mövzular üzrə fikir və mövqeyini rus dilində ardıcıl və əsaslandırılmış şəkildə ifadə edə biləcək. rus dilində akademik və peşəkar məlumatları təhlil, ümumiləşdirmə və təqdimetmə bacarıqlarını nümayiş etdirə biləcək. peşəkar və akademik mühitdə rus dilinin terminoloji və kommunikativ imkanlarından düzgün və məqsədyönlü istifadə edə biləcək.

3. 3

   CLO 3: Methods of numerical solution of the cauchy problem for ordinary differential equations and systems of equations should be given.

4. 4

   CLO 4: For differential equations with ordinary and special derivatives, several methods of solving boundary value problems should be found.

5. 5

   CLO 5: Must know numerical solutions of integral equations.

6. 6

   CLO 6: With the help of computer technologies, he should know the application of numerical methods to the construction of algorithms for solving specific problems.

1. 1.
   Approximate calculation of the integral of a function using the trapezoidal, rectangular and simpson methods

   In the subject, the rules of using numerical integration methods are studied when it is difficult to calculate a definite integral analytically. structure, calculation formulas and application features of rectangular, trapezoidal and simpson methods are explained. also, the comparison of the accuracy of these methods and their implementation in the MATLAB environment are investigated.

   • He should learn the methods of numerical solution of the system of linear algebraic equations, their summation and error estimation, determination of the roots of nonlinear equations and systems of nonlinear equations, finding the roots by simple iteration, bisection, newton and chord methods.
2. 2.
   Solving the cauchy problem for the first order ordinary differential equation by euler's methodLecture

   In the subject, the method of numerical solution of the cauchy problem for the ordinary differential equation of the first order is studied. the basic formula of euler's method, the application of the initial condition and the principle of stepwise approximation are explained. the evaluation of the accuracy of the obtained approximate solution and the execution of calculations in the software environment are also investigated.

3. 3.
   Symmetrical scheme. runge–kutta methods. approximation error of two-stage runge–kutta methodsLecture

   The structure and application principles of symmetric schemes used in the numerical solution of ordinary differential equations are studied in the subject. the general idea of ​​runge–kutta methods is explained, especially the calculation procedure of the two-stage runge–kutta method. also, estimation of the approximation error and accuracy of the two-stage runge–kutta methods is investigated.

   • He should learn the methods of numerical solution of the system of linear algebraic equations, their summation and error estimation, determination of the roots of nonlinear equations and systems of nonlinear equations, finding the roots by simple iteration, bisection, newton and chord methods.
4. 4.
   Multistep finite difference methodsLecture

   The subject studies the essence of multi-step finite difference methods used in the numerical solution of ordinary differential equations. the procedure for calculating the next approximation using the values ​​obtained in the previous few points is explained. also, the effect of these methods on the aggregation, stability and calculation accuracy is investigated.

5. 5.
   Numerical solution of boundary value problem by finite difference method. sweeping methodLecture

   Application of the finite difference method for the numerical solution of boundary value problems is studied in the subject. the replacement of the differential equation with difference equations and the construction of the resulting system of linear equations are explained. also, the efficient solution of tridiagonal systems by the sweeping method is investigated.

6. 6.
   Collocation and galerkin methods.Lecture

   The essence of the collocation and galerkin methods used in the approximate solution of differential and integral equations is studied in the subject. in the collocation method, the solution satisfies the equation at the selected points, and in the galerkin method, the orthogonality of the residual function to the selected basis functions is explained. the use of these methods in the numerical solution of boundary problems and applied mathematical models is also investigated.

7. 7.
   Solving the dirichlet problem for elliptic type differential equations by grid methodsLecture

   Mathematical formulation of the dirichlet problem and boundary conditions for elliptic-type differential equations are studied in the subject. dividing the given domain into a grid, replacing derivatives with finite differences and constructing a difference scheme are explained. the solution of the obtained system of linear equations and the analysis of the numerical results are also investigated.

8. 8.
   Maximum principle. substitution of boundary conditions by various methodsLecture

   In the topic, the essence of the maximum principle for elliptic type differential equations and the evaluation of the behavior of the solution within the domain are studied. the role of the maximum principle in justifying the uniqueness of the solution of border issues is explained. also, rules for replacing boundary conditions with finite difference, interpolation and other numerical methods are investigated.

9. 9.
   Solution of the cauchy problem for differential equations of hyperbolic type using network methodsLecture

   Mathematical formulation and initial conditions of the cauchy problem for hyperbolic type differential equations are studied in the subject. substitution of derivatives by finite differences, construction of the grid domain, and derivation of the appropriate difference scheme are explained. also, the evaluation of the stability and accuracy of the approximate solution obtained by network methods is investigated.

10. 10.
    Solving complex problems for differential equations of hyperbolic type by network methods.Lecture

    Mathematical formulation of mixed problems, initial and boundary conditions for hyperbolic type differential equations are studied in the subject. the replacement of derivatives by finite differences and the construction of a suitable difference scheme in the grid domain are explained. also, the stability, accuracy and compatibility of the obtained approximate solution with boundary conditions are investigated.

11. 11.
    Solving the cauchy problem for parabolic type differential equations using network methods.Lecture

    Mathematical formulation and initial conditions of the cauchy problem for parabolic type differential equations are studied in the subject. the construction of the grid domain, the substitution of derivatives by finite differences, and the derivation of suitable difference schemes are explained. the stability, convergence and accuracy of the approximate solution obtained by network methods are also investigated.

12. 12.
    Solving the dirichlet problem for elliptic type differential equations by the method of straight lines.Lecture

    Numerical solution of the dirichlet problem for elliptic type differential equations by straight lines method is studied in the subject. it is explained that the derivatives of one of the variables in the given area are replaced by finite differences and the problem is brought to the system of ordinary differential equations. the solution of the obtained system and the analysis of the numerical results according to the boundary conditions are also investigated.

13. 13.
    Integral equations. classification of integral equations. method of successive approximations for solving integral equations.Lecture

    In the subject, the concept of integral equation, its connection with differential equations and application areas are studied. fredholm and volterra type integral equations, as well as classification of first and second type integral equations are explained. also, the approximate solution of integral equations by the method of successive approximations is investigated.

14. 14.
    A method of replacing the integral with a finite sum for the solution of integral equations of the fredholm type.Lecture

    In the topic, the method of replacing the integral with a finite sum in the approximate solution of integral equations of the fredholm type is studied. using quadrature formulas, it is explained how to bring an integral equation into a system of linear algebraic equations. also, the solution of the obtained system and the estimation of the accuracy of the approximate result are investigated.

15. 15.
    The method of replacing the kernel with a degenerate kernel for solving integral equations of the fredholm type.Lecture

    In the topic, the method of replacing the kernel with a degenerate kernel in the approximate solution of integral equations of the fredholm type is studied. the reduction of the integral equation to a simpler system of linear algebraic equations by means of a degenerate kernel is explained. the solution of the obtained system and the rules for constructing an approximate solution are also investigated.

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