**Pr. Vagif Ibrahimov**

Baku State University, Azerbaijan

**Talk Title**

**The Determination Of The Intersection Of Some Problems And The Construction Of Efficient Methods To Solve Them**

**Talk Abstract**

The mathematical model for almost of all problems arising in the different industries of natural science and formulating in basically by using differential, integral and integro-differential equations. As is known for investigation of these equations have used the theory of Numerical Integration. And also the application of Numerical Integrating is very wide. Theory of definite integrals with the symmetric boundaries is used in Geometry, Physics, Mechanics and in other subjects of science. It is enough to remind the computing of the energy to transfer signals, calculation of the values of energy receiving in an earthquake for studying the noise in seismology and etc. Therefore, here has defined the intersection of the regions the existence and uniqueness of the solution of initial-value problem for ODE, Volterra integro-differential equations and also the solution of Volterra integral equations by taking into account computing of single and double integrals. By this way have defined the region in which finding the solution of these problems is equivalent.

Thus here investigated the construction of efficient methods and the application of them to solve the above-mentioned four problems. For this aim, here have proposed to use the intersection of the multistep methods with constant coefficients, hybrid, and forward-jumping (advanced) methods. And have been proved that by using the method of unknown coefficients, one can construct stable methods with high exactness having the new properties.

To construct the efficient methods here has proposed the way by using of which the region of the intersection for the existence of the equivalent solution of above-mentioned problems can be extended and estimated the errors received in this case. Let us note that studied here problems have different properties. Therefore here define the basic properties for all considering problems and by using these properties have constructed the special methods for solving the named problems. For example, by comparison of these methods constructed to solve the initial-value problem for the ODE and application of them to calculate the definite integrals have shown that one of the same methods in one case can be taken as the implicit and in other cases as the explicit. As is known the hybrid methods have some advantages and in the application of them to solve the initial-value problem for the ODE arise some difficulties related with the calculation the values of the solution of considering problems at the hybrid points. But in the calculation of the definite integrals, there are not arise any difficulties.

By using this situation here have given some recommendation for the application of the constructed here methods to solve the investigated problems.

Note that to define the values of the order of exactness here are received some relationship by which one can be determined the maximal value for the order of exactness for the stable and unstable methods. These results can be taken as the development of Dahlquist theory. To illustrate, the received results have been constructed some stable methods with the order of exactness and some of them applied to solve the model problems.

**Short Biography**

Ibrahimov is a corresponding member of ANAS and Honored Teacher of the Republic of Azerbaijan. Doctor of Physical and Mathematical Sciences, V.R. Ibrahimov, for investigation of the forward jumping methods, extrapolation and interpolation methods in the general form, has constructed several formulas by which one can determine the upper bound for the accuracy to explicit and implicit stable multistep methods Obreshkov type, such he expanded Dalqvist's theory. For the first time he proved the advantages of the forward jumping methods, and he constructed special methods such as predictor-correction for their use. He proved that there are more precise forward jumping methods. V.R. Ibrahimov found the maximum values of the degrees of stable and unstable MMM (including Cowell type methods) thus the study of the relationship between order and degree for MMM can be considered complete. V. Ibrahimov received a special representation of the error of the multi-step method, with which he determined the maximum number of increase in the accuracy of the method after a single application of Richardson extrapolation and a linear combination of multi-step methods. To construct more precise methods, he proposed using hybrid methods, which he applied to solving first-order and second-order ordinary differential equations. V.R. Ibrahimov defined the relations between of some coefficients for the MMM (including methods with forward jumping), which are the main criterion in the construction of stable multistep methods Obreshkov type with the maximal degree. These relations can be applied to the construction of two-sided methods. It is these methods that allow us to find the interval in which the exact value of the solution of the original problem lies. V.R. Ibrahimov constructed special methods for solving integral equations of Volterra type, in which the number of calculations of the integral kernel at each step remains constant. He defined sufficient conditions for their convergence. Taking into account that these methods represent new directions in the theory of numerical methods for solving integral equations, he constructed methods at the junction of multi-step and hybrid methods applied to solving integral and integro-differential equations of Volterra type. To solve integral equations of Volterra type with symmetric boundaries, he proposed using symmetric methods and constructed special symmetric methods of the forward jumping type. In order to construct stable methods having higher accuracy and an extended stability region, V.R. Ibrahimov proposed to construct methods at the junction of hybrid and forward jumping methods, which applied to the solving of ODE, integral and integro-differential equations of Volterra type. He proved the available to solving of initial-value problem for the ODE and Volterra integro-differential equations and also to solving of Volterra integral equations by taking into account computing of single and double integrals by the one and the same methods. He has estimated the maximal value of the order of exactness all the proposed methods.

Awards

2014- Diploma awarded by the Foundation for the Development of Science under the President of the Republic of Azerbaijan, the Ministry of Communications and High Technologies of the Republic of Azerbaijan and the State Commission of the Republic of Azerbaijan by UNESCO (awarded second place for the best work in the field of ICT).

2011-2014 -Grant issued by the Foundation for the Development of Science under the President of the Republic of Azerbaijan .

2016-2019 -Grant issued by the Foundation for the Development of Science under the President of the Republic of Azerbaijan .

2011 - Diploma "Development of Science", issued by the international organization ASHE .

2009 - Honored Teacher of the Republic of Azerbaijan.

**Talk Keywords**

**Target Audience**

**Speaker-intro video**