Mehriban Imanova

Baku State University | BSU · Computational Mathematics

The many problems of natural sciences are reduced to solving integro-differential equations with variable boundaries. It is known that Vito Volterra, for the study of the memory of Earth, has constructed the integro-differential equations. As is known, there is a class of analytical and numerical methods for solving the Volterra integro-differentia...

Mathematical models for many theoretical and practical problems of natural sciences have been described by the ODEs. Such processes are common in the investigation of physical processes, which the scientists have begun to study from Newton (it is enough to remember Newton's laws). In recent times it often arises the necessity to construct mathemati...

As is known one of the well-researched mathematical problems is the initial-value problem for the ODEs. There are some classes of methods for solving this problem. The present time has been constructed the new classes of methods of advanced and hybrid types. By using these methods have been constructed the class of methods on the intersection of th...

The scientists began investigate of the solution of ODE from the XVII century. Many famous mathematicians as Newton, Leibniz, Bernoulli, D’ Alembert, Euler, Cauchy and etc. have considered solving of the ordinary differential equations. For solving these equations the scientists from the different country have constructed many methods. Here, with t...

Dear friends and colleagues, it is a pleasure for me to inform you about visiting of Dr. Muhammad Sharif who is an Advisor Science and Technology at Islamic World Educational, Scientific and Cultural Organization (ICESCO), Rabat, Kingdom of Morocco. Prior to join ICESCO Dr. Sharif was serving as Asst. Professor at King Fahd University of Petroleum...

There are some classes of methods for solving integral equations of the variable boundaries. It is known that each method has its own advantages and disadvantages. By taking into account the disadvantages of known methods, here was constructed a new method free from them. For this, we have used multistep methods of advanced and hybrid types for the...

It is known that the mathematical models of different industries of the natural sciences reduce to the solving of initial-value problem for ODEs of the second order. To solve this problem often used the finite-different methods. For this aim, here have proposed to use the multistep methods with constant coefficients and also multistep hybrid method...

It is known that one of the popular methods constructed to solve the scientific and engineering problem is the finite difference method studied beginning from the known and famous scientists as the Newton, Leibniz, Euler and etc. One of the first applications of the finite difference method is defined as the conception of the derivatives for the co...

As is known there is the wide class of methods for calculation of the definite integrals constructed by the well-known scientists as Newton, Gauss, Chebyshev, Cotes, Simpson, Krylov and etc. It seems that to receive a new result in this area is impossible. The aim of this work is the applied some general form of hybrid methods to computation of def...

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...

As is known there is the wide class of methods for calculation of the definite integrals constructed by the well-known scientists as Newton, Gauss, Chebyshev, Cotes, Simpson, Krylov and etc. It seems that to receive a new result in this area is impossible. The aim of this work is the applied some general form of hybrid methods to computation of def...

As is known the necessity to solve the initial-value problem for the Volterra integro-differential equations arises in investigation of the residual knowledge of some objects. Volterra by using the integro-differential equations has studied the memory of land and many other important problems which have been arisen in the study of some phenomena fr...

ODE and its applications are studying very long ago.And therefor, there are wide classes of methods for solving these equations, which are fundamentally investigated by many autors.Here by using some relation between the Volterra integral equations and ODE have constructed the hybirid multistep methods of the forward jumping type, to solving the Vo...

It is known that in the construction of the numerical methods for solving of the initial-value problem of ODE in basically used the methods which have applied to the calculation of the definite integrals. Here for the computing of definite integrals propose to use the methods which have used in solving of the initial-value problem for the ODEs. The...

As is known, there are some classes of numerical methods for solving of the initial-value problem for the Volterra integro-differential equations. Here, by comparison of the known methods have constructed the methods with the new properties which have applied to solve the initial-value problem for the ODE and for the Volterra integra-differential e...

There are some wide classes of the numerical methods to solving ODE. Therefore the specialists tried to apply these methods to solving of the Volterra integral and integro-differential equations. Here, we proposed the new relation between ODEs and Volterraintegro-differential equation, by using which have constructed the stable multistep methods, h...

There are some classes of the methods for solving the initial-value problem of the integro-differential equations of Volterra type. Obviously, each class of these methods has its advantages and disadvantages. Therefore, here we consider a comparison of some known methods by which the region of the application of the certain methods is determined. T...

As is known, in the study of many problems of natural science there arises the need to find numerical solutions of the initial value problem for the Volterra integro-differential equations. Taking into account that the solution of the initial value problem for ODEs has been fundamentally investigated by many authors, the specialists try to apply th...

There is a wide range of numerical methods for solving differential equations. Each method has advantages and disadvantages. Criteria to evaluate these methods include stability, highest degree, an extended stability region, simple structure, etc. Our numerical solution for ordinary differential equations, the second derivative hybrid method, is co...

One of the main problems in the theory of numerical methods is the construction of the methods with the high accuracy and having extended stability regions. Depending from the object of investigation, it usually requires the necessary of some additional requirements on the using methods as a decreasing ofvolume computational work, the using of the...

As is known, the model problem for some scientific and technical problems is formulated with the help of the integro-differential equations of Volterra type of the second order. And for solving of such equations, numerical methods are mainly used. Numerical solution of nonlinear integro-differential equations of Volterra type has been studied relat...

As is known there are the mainly two classes of the numerical methods for solving ODE, which is commonly called a one and multistep methods. Each of these methods has certain advantages and disadvantages. It is obvious that the method which has better properties of these methods should be constructed at the junction of them. In the middle of the XX...

It is known that solving of many scientific and application problems can be regarded as the solving of integral equations with the variable boundaries. Among all the integral equations, the most popular ones are those in which one of the boundaries of the integral is fixed. Here, we investigate one particular case in which both boundaries of the in...

As is known, the theory of integral calculations used in almost all the fields of the natural science, as computes the volume of rotating bodies, areas having different shapes, distances between some objects, and etc. The last time exploring the energy signals are studying earthquake, distribution of telecommunications signals, etc. Among these, th...

It is well known that the study of many processes of the natural sciences can be reduced to solving integro-differential equations with variable boundaries. Recently, studies on certain problems of the environment, such as the flu virus, the emergence of new diseases, and diseases associated with mutations of viruses, have become relevant. A soluti...

There is a wide range of numerical methods for solving differential equations. Each method has advantages and disadvantages. Criteria to evaluate these methods include stability, highest degree, an extended stability region, simple structure, etc. Our numerical solution for ordinary differential equations, the second derivative hybrid method, is co...

In nature, we are often faced with the damped phenomena. It is known that some of them continue to exist for a long period of time, and some contrary have completed their life journey over the shortest period of time. Typically, the study of such phenomena applies on the theory of integral equations with the symmetric bounders. Here, are considered...

As is known, the solution of some problems of ecology, geophysics, nuclear physics, the study of some seasonal distribution of the disease and so on, reduced to solving of integro-differential equations of higher order. Note that solving of these equations can be reduced to solving system of integro-differential equations of the first order. Howeve...

In the middle of XX century the scholars began construct the methods which have the best characteristics of the known and multi-step methods with the constant coefficients. And in the 1955 years there appears the works of Gear and Butcher dedicated investigation of the hybrid method. Here we want to show how this theory developed and compared them...

The classic approach at solving integral equations with the variable boundaries consists of some methods of the theory of difference methods or differential equations. In contrast from the approach here are considered the relationship between methods of application to the solution of differential equations and to the solution of integral equations...

In the middle of XX century the scholars began construct the methods which have the best characteristics of the one and multi-step methods with the constant coefficients. And in the 1955 years there appears the works of Gear and Butcher dedicated investigation of the hybrid method. Here we want to show that how this theory developed and compared th...

Since ancient times, people mainly use the concept of symmetry for the approval of the greatness of the creator, to illustrate of the beauty of the object or of the approval the talent. Therefore, the concept of symmetry can be considered as a well-known notation for all people. Obviously, for the showing of the symmetry of the object it is not nec...

For solving ordinary differential equations there is a wide arsenal of numerical methods, among which of them the most popular are the methods of the Runge- Kutta and Adams. Currently, these methods have become classics and new directions in the theory of numerical methods, such as Obreshkov methods, methods of forward-jumping type, hybrid methods...

The scientists began investigate of the solution of ODE from the XVII century. Many famous mathematicians as Newton, Leibniz, Bernoulli, D' Alembert, Euler, Cauchy and etc. have considered solving of the ordinary differential equations. For solving these equations the scientists from the different country have constructed many methods. Here, with t...

One of priorities direction in numerical mathematics is the investigation of the numerical solution of integro-differential equations. As is known, many vital tasks such as research in the field of atomic physics, ecology, geophysics, to extended infectious diseases and, etc. reduced to solving of integro-differential equations. Here, applied forwa...

As is known, in solving many scientific and applied problems are using the hybrid methods that have investigated in the scientific world about 50 years. During these years, the study of hybrid methods have moved into a new phase, where they began to be applied them to the solution of integral and integro-differential equations. In contrast to these...

Solving of integro-differential equations with variable boundaries encounters in many areas of natural science. Numerous articles and monographs are published on the qualitative theory and numerical methods for solving such equations. However, among these works there are almost no papers on the determination of the stability of numerical methods. T...

Scientists basically began studies of the numerical solution of the Volterra integral equation after publishing of the known paper of Volterra. To define solution of such equation they used quadrature methods. The first quadrature method for solving linear integral equations with the variable boundaries has been constructed by Volterra. For receivi...

As is known, solving of some problems in the ecosystem, communications, computational biology, mechanics and etc. are reduced to the computation of symmetric integrals. In some cases, traditional methods are applied to the calculation of these integrals, not keeping the suitable results. Therefore appears a need to constructing special methods for...

Mathematical models for solving many practical problems is formulated in the form of integral equations. But find the solutions of these equations, even in the linear case is not possible always. Therefore to solving of integral equations are applied approximate methods , among which the most popular is the quadrature methods. In the last time cons...

Recently, many scientists have focused on the investigation of variable boundary integral equations that describe some phenomena of natural science, such as environmental problems, ecological problems, and the spreading of grippe and other seasonal diseases. The method of quadratures that Volterra first used to solve variable boundary linear integr...

Constructed hybrid methods of the high accuracy the experts examined that’s for solving integral and integro-differential equations. Using hybrid methods for solving integral equations belongs to Makroglou. Here, developing these idea, explored a more general hybrid method which is applied to solving Volterra integral equations and also constructed...

There are wide classes of numerical methods for solving ordinary differential equations, which are involved in the requisite models for almost all of the problems of natural science. To compare these methods, scientists have proposed using some of the following criteria: stability, exactness, and stability regions. As a result, both the theoretical...

As it is known, the solution of many problems of natural science can be reduced to the solution of ordinary differential equations of the first and second orders. Typically solutions for ODEs second order are reduced to solving a system for differential equations of first order. Here, has been analyzed known numerical methods for chronological mann...

It is known that to construct the stable multistep method with the higher order of accuracy for solving integral equation is actual. For this aim here we suggest some ways for the construction of hybrid methods for solving nonlinear Volterra integral equations of the second kind. Thus, foundational this extends stable hybrid method with higher orde...

It is well known that in the early XXth century, to solve some problems in the field of mechanics Vito Volterra had to solve integro-differential equations with variable boundaries. Several authors studies integro-differential equations in junction of differential and integral equations. Therefore they used quadrature method or its modification. Am...

Розглянуто побудову методів розв’язання звичайних диференціальних рівнянь другого порядку, а саме гібридних методів. З цією метою побудовано конкретні методи зі ступенем , а також запропоновано алгоритм для їх реалізації.

As is known, determination the solution of many problems of natural science can be reduced to determination of the solution of first and second orders ordinary differential equations. Typically solutions for ODEs second order are reduced to solving a system differential equations of first order. Here, has been analyzed known numerical methods for c...

It is known, that solution of many problems of modern mathematics reduces to solving Volterra integral equations. One of the popular methods for solving Volterra integral equation is the quadrature method. To the increases order of accuracy for the multistep methods some authors suggested some modification then and using hybrid methods. Here, for c...

As is well known investigation of many processes of natural sciences reduce to the solving of initial value problem for integro-differential equations which are one of the priority areas of modern mathematics. To define the exact solution of such problems is not always possible. Therefore the scientists constructed approximate methods for solving t...

While constructing mathematical model of some phenomena of nature we meet the solution of initial value problem for Volterra integro-differential equation. Many papers have been devoted to the numerical solution of this problem. In these works, at first the integral is replaced by the integral sum then one of the well investigated numerical methods...

While constructing mathematical model of some phenomena of nature we meet the solution of initial value problem for Volterra integro-differential equation. Many papers have been devoted to the numerical solution of this problem. In these works, at first the integral is replaced by the integral sum then one of the well investigated numerical methods...

Starting with Volterra's paper, the scientists investigate the solutions of scientific-technical problems by means of integro-differential equations. For finding the approximate solutions of such equation, the quadratures method suggested by Volterra were mainly used. Taking into account some advantages of hybrid methods Makroglou applied them to n...

With the numerical solution of ordinary differential equations(ODE), scientists engaged in the Middle Ages, beginning with the work of Clairaut. The domain of the numerical methods involved in many famous mathematicians - Euler, Runge, Kutta, Adams, Laplace, and others. They have constructed methods with different properties. In this paper we consi...

As is known, many problems of natural science are reduced mainly to the solution of nonlinear Volterra integral equations. The method of quadratures that was first applied by Volterra to solving variable boundary integral equations is popular among numerical methods for the solution of such equations. At present, there are different modifications o...

As is known, many problems of natural science are reduced mainly to the solution of nonlinear Volterra integral equations. The method of quadratures that was first applied by Volterra to solving variable boundary integral equations is popular among numerical methods for the solution of such equations. At present, there are different modifications o...

As is known, one of the priority directions of research works of natural sciences is introduction of applied section of contemporary mathematics as approximate and numerical methods to solving integral equation into practice. We fare with the solving of integral equation while studying many phenomena of nature to whose numerically solving by the me...

It is known that having existed on earth humanity tried to study and further all the processes happening around him. Therefore, starting with ancient times the scientists studied the man himself, his connections with world around, his relation to environment, activity of men on the whole. However, in the represented paper the object of investigatio...

Taking into account that many problems of natural sciences and engineering are reduced to solving initial-value problem for ordinary differential equations, beginning from Newton, the scientists investigate approximate solution of ordinary differential equations. There are papers of different authors devoted to the solution of initial value problem...

There exists a wide class of numerical methods for solving of initial-value problem for ordinary differential equations. These classes may be divided into onestep and multistep ones. For constructing hybrid methods, some relations between one and multistep methods are researched on the base of Runge-Kutta and Adams methods. By means of Runge-Kutta...

Considering wide application of integro-differential equation, last time scientists from different countries have actively engaged in research of approximate solutions of integrodifferential equations. Many of these scientists mainly are engaged in numerical solution of Volterra integro-differential equations. For this purpose, quadratic method, mo...

Considering wide application of integro-differential equation, last time scientists from different countries have actively engaged in research of approximate solutions of integrodifferential equations. Many of these scientists mainly are engaged in numerical solution of Volterra integro-differential equations. For this purpose, quadratic method, mo...

Solution of some practical problems is reduced to the solution of the integro-differential equations. But for the numerical solution of such equations basically quadrature methods or its combination with multistep or one-step methods are used. The quadrature methods basically is applied to calculation of the integral participating in right hand sid...

While researching some applied problems we face with a solution of the Cauchy problem for integro-differential equations. As is known, by solving integro-differential equations we widely use approximate methods. These methods, commonly, have two ways of the development: one of them in reduction of integro-differential equations to differential equa...

At solution of many applied problems we collide with the solution of the integro-differential equations. Among these problems we can note the population of biological objects. One of the first works devoted to the solution of the indicated problems is Volterra’s work (see V. Volterra [Remarques sur la Note de M. Regnier et M-lle. Lambin., C. R. 199...