Author: Allaberen Ashyralyev
Born: 01 August 1955
Bio: Prof. Ashyralyev graduated from Kara-Kala, Turkmenistan N5 high school with a gold medal in 1972.  He became a student at the Turkmen State University in 1972. From 1974 to 1977 he had been awarded by Lenin’s Grant for students. He graduated from this University with the highest rank in 1977. Afterwards he became a teacher at the Department of Mathematical Analysis at the Turkmen State University.
In 1979, he participated in Professional Development Program at the Voronezh State University, Russia. There he developed a strong interest in mathematics. In 1980, he became a PhD student at the Department of Functional Analysis and Operator Equations at the Voronezh State University, Russia. For successful achievements in his research, Ashyralyev had been awarded with the First Rank Prize for Ph.D students in 1983. The same year he obtained his Ph.D degree at the Voronezh State University with the highest rank. 
In his Ph.D work, Ashyralyev had shown for the first time that the theory of interpolation operators in a Banach space is very important in investigations of the stability and coercive stability of difference schemes for parabolic and elliptic equations. This discovery was highlighted by some other scientists in this field. 
Prof. Ashyralyev, a scholar from the USSR, started his research career in 1987 when he got admission to the Department of Mathematics for Professional Development Exchange Program at the University of California, Santa Barbara, USA to work on a project entitled ”Theory of positive operators and investigation of difference schemes”. During his stay in USA, Ashyralyev had shown for the first time that the theory of positive operators in a Banach space is very important in investigations of the theory of stability of difference schemes for PDEs and stochastic parabolic equations. In particular, new exact single-step difference scheme for the first order evolution equations with arbitrary positive parameter in the high derivative and two-step exact difference schemes for parabolic and elliptic equations with arbitrary positive parameter in the high derivative were presented. Applying these exact difference schemes, Ashyralyev constructed and investigated a high order of accuracy uniform difference schemes with fitting operator for the approximate solution of several perturbation problems. This discovery was highlighted by some other scientists in this field. Results of his PhD work and research done during Professional Development Exachange Program in USA were published in more than 10 papers in very high quality journals in mathematics, including Dokl. Akad. Nauk SSSR, Dokl. Akad. Nauk Ukrainian SSR, Ser. À Fiz.-math. and Tech. Sciences, Numerical Functional Analysis and  Optimization, Applicable Analysis and SIAM J. Math. Analysis.
        Professor Ashyralyev completed his second PhD Degree (Doctor of Sciences in Mathematics) at the Functional Analysis and Operator Equations Department of Russian Voronezh State University and Mathematics Institute of Ukraine Science Academy in 1992. He continued his career at the Turkmen State University from 1977 to 1999 as instructor, assistant professor (1985), associate professor (1989), and professor (1995), respectively. Since 1999 Prof. Ashyralyev is a staff member of the Department of Mathematics, Fatih University and is a joint professor at the International Turkmen-Turk University, Ashgabat, Turkmenistan.  His research fields are the theory of ordinary and partial differential equations, stochastic partial differential equations, numerical analysis, computational mathematics, numerical functional analysis and their applications. In particular, his scientific interests include: well-posedness of differential and difference problems, construction and investigation of high order of accuracy difference schemes for partial differential equations, uniform difference schemes and asymptotic formulas for singular perturbation problems for partial differential equations, mathematical modeling, study of positivity of differential and difference operators and structure of fractional spaces generated by positive differential and difference operators in Banach spaces. He is the author of more than 250 scientific papers in international journals and conference proceedings and 10 books including two monographs published by Birkhauser-Verlag, in Operator Theory: Advances and Applications.

 He has been the member of the advisory board of a number of national and international mathematics conferences and workshops. He is the editor of 10 international journals besides Mathematical Reviewers and referee of  30 national and international qualitative journals. 

In 2009, Professor Ashyralyev was selected as the most succesful national scientist of Turkmenistan in last 30 years (second rank) by MyNetResearch Empowering Collaboration Search (which is based on the papers published in journals in Web of Science).

Professor Ashyralyev is best known for his two main contributions:

1. The monograph Ashyralyev A. and Sobolevskii P.E., Well-Posedness of Difference Parabolic Equations, Birkhauser Verlag: Basel. Boston. Berlin, vol.69, 1994, 349 pages. 
Mathematical models described by various real dynamic processes can reduce to boundary value problems for partial differential equations. A well-known widely applied method of approximate solutions of various problems of partial differential equations is the method of difference schemes. The main characteristics of difference schemes are their accuracy and stability.  Modern computers allow us the implementation of highly accurate difference schemes. Hence, a task of current interest is the construction and investigation of highly accurate difference schemes for various boundary value problems for partial differential equations. 
This monograph is devoted to the construction of highly accurate difference schemes for parabolic boundary value problems, based on Pade approximations. The investigation is based on a new notion of positivity of difference operators in Banach spaces, which allows one to deal with difference schemes of arbitrary order of accuracy. Establishing coercivity inequalities allows one to obtain sharp, that is, two-sided estimates of convergence rates. The proofs are based on results in interpolation theory of linear operators. This approach  is important to develop the theory of difference schemes  of high order accuracy for various boundary value problems for partial  differential equations which  resulted  in  the  important  scientific  discoveries  in  this  field.  Nowadays, many  mathematicians  apply  this  approach  to  investigate  various problems of  mathematical  physics  and   stochastic   differential  equations  which  appear  in  applied  problems.  

2. The monograph Ashyralyev A. and Sobolevskii P. E., New Difference Schemes for Partial Differential Equations,  Birkhauser Verlag: Basel. Boston. Berlin, 2004, vol.148, 443 pages. This monograph is devoted to the construction and investigation of the new high order of accuracy difference schemes of approximating the solutions of regular and singular perturbation boundary value problems for partial differential equations. The construction is based on the exact difference scheme and Taylor's decomposition on the two or three points. This approach permitted essentially to extend a class of problems where the theory of difference methods is applicable. Namely, now it is possible to investigate the differential equations with variable coefficients and regular and singular perturbation boundary value problems. The investigation is based on new coercivity inequalities.

One of the most important aspects of Dr. Ashyralyev's work is that he has shown in his studies that his mathematical results obtained theoretically are useful for solving various real-life problems. For example, the investigation of the amount of underground natural gas sources is one of the most important problems in Turkmenistan.  
The solutions of large-scale scientific-technological problems in the field of construction of the base of project and system of rational elaboration, improvement of elaboration and exploitation of technology, modern methods of construction deep well-hole in complicated conditions, hydrodynamics of low permeable spaces, protection of enviroment became possible only due to application of mathematical models and new numerical methods implemented on computers. Mathematical models of many of such type problems are reduced to nonclassical or classical problems of equations of mathematical physics.
 Dr. Ashyralyev jointly with Prof. A.N. Muradov and Prof. S. Batyrov from Turkmen Polytechnic Institute and their group of scientists had been investigating the modeling processes of exploitation of gas places. The investigation of the underground natural gas 6 km beneath the earth surface is based on the mathematical models. Unfortunately, it is inverse and ill-posed problem and it is not possible to obtain its exact solution. Ashyralyev used the stable numerical methods for the solution of these problems. It was an important numerical algorithm for calculation of the amount of natural gas 6 km beneath the earth surface. Dr. Ashyralyev has obtained stable difference methods for solution of some problems arising in this field.
The main merit of Ashyralyev's work is that he has investigated the theory of difference schemes independently on classical approaches (maximum principle or others). Moreover, it was a starting point for the investigation of Ashyralyev's work in PDE.

The application of the solar energy for the obtaining of solution of various problems in the nature is one of the important problems in Turkmenistan.  
Solar Energy Institute in Academy of Sciences in Türkmenistan was the only single institute of the Academy of Sciences of the Soviet Union in the Soviet time. Various investigations were made to use solar energy for daily life. Dr. Ashyralyev and Professor L.E. Ribakova from Turkmen State University had a project on the winter greenhouse examined the establishment of the optimal geometry. Ashyralyev used the stable numerical methods for the solution of these problems. It was an important numerical algorithm for the solution of the problem arising in the winter greenhouse. Moreover it was a starting point for the investigation of Ashyralyev's work in partial differential equations with nonlocal boundary conditions. In present, Dr. Ashyralyev jointly with his PhD students studies various nonclassical problems for PDE, arising in applied sciences.
Since 1999, during his stay in Turkey he had joint work with scientists from engineering departments for solving the various problems arising in applied sciences and in the nature. For example, Dr. Ashyralyev jointly with BS student Doghonay Arjmand from  Deparment of Electrical and Electronic Engineering of the Bogazici University  and  Prof. M. Koksal from Deparment of Electrical and Electronic Engineering of the Fatih University, Turkey presented the three-step difference schemes for the numerical solutions of third-order time-varying linear differential equations. The construction of the three-step difference schemes is based on Taylor’s decomposition on four points. The results are shown to be well applicable for the numerical solution of nonlocal boundary-value problems by an example involving periodically time-varying parameters. This result is important in communication systems with many subsystems such that usefully modeled as linear periodically time-varying systems.  Moreover it was also a good starting point for the investigation of Ashyralyev's work in partial differential equations with nonlocal boundary conditions.
Another important study done by Dr. Ashyralyev jointly with A.S. Erdogan and Prof. N. Arslan from  Deparment of Genetics and Bioengineering  of the Fatih University, Turkey was Atherosclerosis (swelling of the vascular periphery of the cells, lipids, or calcium accumulation due to reasons such as depletion of atherosclerosis or hardening of the arteries), a cause of much disease and death. For example, according to a study conducted in USA in 1992 nearly a million people died due to these reasons. This rate is two times higher than deaths from cancer and ten times higher than the deaths in traffic accidents. Despite major advances in medicine in recent years, observed coronary artery disease (causing a heart attack and result of atherosclerosisin) and blockage of the arteries in the brain due to atherosclerosis, rupture, or bleeding caused more deaths than by other diseases. Atherosclerosis affects large and medium-sized vessels. Atherosclerosis is a ‘slow’ disease which may develop from early childhood. Computational blood flow analysis through glycocalyx located over the endothelial cells in human capillary vascular system has been performed. Dr. Ashyralyev jointly with A.S. Erdogan and Prof. N. Arslan presented stable numerical schemes for obtaining approximate solutions to the mixed problem to the partial differential equation with variable space operator to model the blood flow. Numerical techniques were developed by applying a procedure of the solution to the linear difference equation with matrix coefficients. The flow equations inside the core flow region and porous region were established. Discretization was done and the flow velocities in both regions were calculated. The effect of the flow over the glycocalyx was investigated. The wall shear stresses and the drag force over the glycocalyx were calculated. These results seem to be very promising and may serve as a starting point for other scientists in the future. 

Dr. Ashyralyev’s contribution in human resource development has been remarkable.  He has guided 19 PhD students (8 as a supervisor + 11 as a committee member). In addition, he supervised 16 MS and 30 BS graduates.  He has also supervised a number of PhD students in Turkmenistan; their work is almost in completion stage. 
Dr. Ashyralyev's contribution to science and technology within Turkmenistan and Turkey has been tremendous which could be gauged from his active participation in a number of national research and academic organizations. Furthermore, his contribution in science and technology at the international level is equally remarkable which is evident from his current joint research with scientists in different countries including USA(jointly with Fattorini, H.O., San, M.E. and Agarwal, R.P.), Germany (jointly with Weis, L.), Saudi Arabia(jointly with Akca, H.), Russia(jointly with Sobolevskii, P.E. and Piskarev, S.), Israel(jointly with Sobolevskii, P.E.), Poland(jointly with Biszewski, L.), Spain(jointly with Martinez, M. and Pastor, J.), Brazil(jointly with Claudio, Cuevas ) and Hungary(jointly with Micheletzky,G.).