**International Conference on Innovative Applied Energy**** **

**E-Proceedings ISBN: 978-1-912532-05-6**

St Cross College, University of Oxford, United Kingdom

## On the Construction of the Multistep Methods to Solving the Initial-Value Problem for ODE and the Volterra Integro-Differential Equations

**Vagif Ibrahimov, Galina Mehdiyeva and Mehriban Imanova**

Department of Computational MathematicsBaku State University, Azerbaijan

**Paper Abstract**

ODE and its applications are studying very long ago. And therefore, there are wide classes of methods for solving these equations, which are fundamentally investigated by many authors. Here by using some relation between the Volterra integral equation and ODE have constructed the hybrid multistep methods of the forward jumping type, to solving the Volterra integral equations, which are the same with the methods have applied to solving ODE. By using methods of undetermined coefficients have received the system of linear and nonlinear algebraic equations for determine the values of the coefficients of these methods. As is known by using the solutions of these systems one can be determined the order of exactness of these methods.In some cases, we succeeded have estimated the values of the order of exactness for our methods And also the necessary conditions imposed on the coefficients of the proposed method have been studied. The methods, constructed here, are applied to solving Volterra integro-differential equations, which are the same with the methods, which are applied to solving ODE. Constructed concrete methods with the order of exactness p<=9 some of which have been applied to solving model problem.

**Paper Keywords**

**Corresponding author 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 ordinary differential equation and the Volterra integral equation by the same 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. Short biography of the keynote speaker.