# Erkinjon Tulkinovich KarimovFergana State University · Mathematical Analysis and Differential Equations

Erkinjon Tulkinovich Karimov

DSc in Mathematics and Physics

## About

91

Publications

25,351

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849

Citations

Citations since 2017

Introduction

Additional affiliations

January 2020 - May 2020

February 2019 - May 2019

September 2018 - January 2019

Education

November 2002 - November 2005

**Institute of Mathematics, Uzbekistan Academy of Sciences**

Field of study

- Differential equations

September 1993 - July 1998

**Fergana State University**

Field of study

- Mathematics

## Publications

Publications (91)

We aim to study Mittag-Leffler type functions of two variables D 1 (x, y) , ..., D 5 (x, y) by analogy with the Appel hypergeometric functions of two variables,. Moreover, we targeted functions E 1 (x, y) , ..., E 10 (x, y) as limiting cases of the functions D 1 (x, y) , ..., D 5 (x, y) and studied certain properties, as well. Following Horn's meth...

We aim to study a unique solvability of a boundary-value problem for a time-fractional diffusion equation involving the Prabhakar fractional derivative in a Caputo sense in a bounded domain. We use the method of separation of variables and in time-variable, we obtain the Cauchy problem for a fractional differential equation with the Prabhakar deriv...

In this paper, we have introduced the Prabhakar fractional q-integral and q-differential operators. We first study the semi-group property of the Prab-hakar fractional q-integral operator, which allowed us to introduce the corresponding q-differential operator. Formulas for compositions of q-integral and q-differential operators are also presented....

In this paper, we explore the weak solution of a time-dependent inverse source problem and inverse initial problem for the q-analog of the heat equation. As an over-determination condition we have used an integral type condition on the space variable (in the case of an inverse source problem) and final time condition (in the case of an inverse init...

The aim of this paper is the investigation of the existence and uniqueness of solutions to Cauchy-type problems for fractional q-difference equations with the bi-ordinal Hilfer fractional q-derivative which is an extension of the Hilfer fractional q-derivative. An approach is based on the equivalence of the nonlinear Cauchy-type problem with a nonl...

A boundary value problem with a nonlocal m-point condition in time for a space-degenerate partial differential equation involving the bi-ordinal Hilfer fractional derivative is the main subject of the present investigation. We aim to prove a unique solvability of this problem based on certain properties of the Legendre polynomials and the two-param...

In a rectangular domain, a boundary‐value problem is considered for a mixed equation with a regularized Caputo‐like counterpart of hyper‐Bessel differential operator and the bi‐ordinal Hilfer's fractional derivative. By using the method of separation of variables a unique solvability of the considered problem has been established. Moreover, we have...

A boundary value problem with a nonlocal m-point condition in time for a space-degenerate partial differential equation involving the bi-ordinal Hilfer fractional derivative is the main subject of the present investigation. We aim to prove a unique solvability of this problem based on certain properties of the Legendre polynomials and the two-param...

A boundary value problem with a nonlocal m-point condition in time for a space-degenerate partial differential equation involving the bi-ordinal Hilfer fractional derivative is the main subject of the present investigation. We aim to prove a unique solvability of this problem based on certain properties of the Legendre polynomials and the two-param...

Metrik graflarda Hilfer operatori qatnashgan vaqt bo'yicha kasr tartibli differensial tenglama uchun masalaning yechilishi Ushbu maqolada biz yulduz ko'rinishidagi metrik grafda Hilfer operatori qatnashgan vaqt bo'yicha kasr tartibli differentsial tenglama uchun bir lokal masalani o'rganamiz. O'zgaruvchilarni ajratish usulidan foydalanib, biz o'rga...

In this paper, we have considered two different sub-diffusion equations involving Hilfer, hyper-Bessel and Erdelyi-Kober fractional derivatives. Using a special transformation, we equivalently reduce the considered boundary value problems for fractional partial differential equation to the corresponding problem for ordinary differential equation. A...

In this paper, we consider a non-local boundary-value problem for a mixed-type equation involving the bi-ordinal Hilfer fractional derivative in rectangular domain. The main target of this work is to analyze the uniqueness and the existence of the solution of the considered problem by means of eigenfunctions. Moreover, we construct the solution of...

In this paper, we consider a nonlocal boundary-value problem for a mixed-type equation involving the bi-ordinal Hilfer fractional derivative in a rectangular domain. The main target of this work is to analyze the uniqueness and the existence of the solution of the considered problem by means of eigenfunctions. Moreover, we construct the solution of...

The issues of unique solvability of a boundary value problem for a mixed type integro-differential equation with two Caputo time-fractional operators and spectral parameters are considered. A mixed type integro-differential equation is a partial integro-differential equation of fractional order in both positive and negative parts of multidimensiona...

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann-Liouville derivatives is less understood. In this paper, we propose new type initial, inner, and inner-boundary value problems for fractional differential equations with the Riemann...

In a rectangular domain, a boundary-value problem is considered for a mixed-type equation with a regularized Caputo-like counterpart of hyper-Bessel differential operator and the double-order Hilfer fractional derivative. Using the method of separation of variables, Laplace transform, a unique solvability of the considered problem has been establis...

The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differential equation with respect to the main unknown function is an inhomogeneous partial integro-differe...

In this work, we investigate Frankl-type problem with integral conjugating condition for a mixed type equation consisting of sub-diffusion and wave equations. A uniqueness and the existence of formulated problem have been proved using energy integrals and the method of integral equations, imposing certain conditions on given data.

Ghent Analysis & PDE Center is organising the International Workshop on Fractional Calculus, 9-10 June, 2020. The scientific program will consist of 60-, 45- and 30-minute lectures. Website https://analysis-pde.org/workshop-fractional-calculus

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann-Liouville derivatives is less understood. In this paper we propose new type initial, inner and inner-boundary value problems for fractional differential equations with the Riemann-L...

In this paper, we have considered time-fractional wave equation with the Riemann-Liouville and Atangana-
Baleanu fractional derivatives. For the first time, solution of the Cauchy problem for fractional differential
equation with the Riemann-Liouville and the Atangana-Baleanu derivative has been found in an explicit form
(See (18)) under certain co...

A unique solvability of the Tricomi type problem with integral conjugating condition for a mixed type equation with hyper-bessel fractional operator has been proved. Main tools are method of energy integrals and integral equations. Considered mixed domain consists of rectangular and characteristic triangle.

In this paper, we have considered a Tricomi type problem for a mixed type equation with Hilfer's double order derivative sub-diffusion equation and classical wave equation in a composite domain. The main methods of the investigation are the method of integral equations and the energy integrals' method.

In this paper, we have considered a Tricomi type problem for mixed type equation with sub-diffusion equation with Hilfer's double order derivative and classical wave equation in a composite domain. Main tools of the investigation are a method of integral equations and energy integrals' method.

In this paper, we investigate two inverse source problems for degenerate time-fractional partial differential equation in rectangular domains. The first problem involve a space-degenerate partial differential equation and the second one involve a time-degenerate partial differential equation. Solutions to both problem are expressed in series expans...

In this work, The Tricomi type boundary problem with integral conjugation condition on the type-changing line for the mixed type equation with Hilfer fractional differential operator has been considered. Using the method of integral equations, energy integral's method, a unique solvability of the formulated problem has been proved.

In this work, The Tricomi type boundary problem with integral conjugation condition on the type-changing line for the mixed type equation with Hilfer fractional differential operator has been considered. Using method of integral equations, energy integral's method, a unique solvability of the formulated problem has been proved.

In the present work, an initial value problem involving the Atangana-Baleanu derivative is considered. An explicit solution of the given problem in integral form is obtained by using the Laplace transform. The use of the given initial value problem is illustrated by considering a boundary value problem in which the solution is expressed in the form...

Direct and inverse source problems of a fractional diffusion equation with regularized
Caputo-like counterpart hyper-Bessel operator are considered. Solutions to these
problems are constructed based on appropriate eigenfunction expansion and results on
existence and uniqueness are established. To solve the resultant equations, a solution to a
non-h...

In the present work, we discuss the existence of a unique positive solution of a boundary value problem for nonlinear fractional order equation with singularity. Precisely, order of equation $D_{0+}^\alpha u(t)=f(t,u(t))$ belongs to $(3,4]$ and $f$ has a singularity at $t=0$ and as a boundary conditions we use $u(0)=u(1)=u'(0)=u'(1)=0$. Using fixed...

In this paper, we investigate direct and inverse source problems for the diffusion equation with two-term generalized fractional derivative (Hilfer derivative) in a rectangular domain. Using spectral expansion method, we derive two-term fractional differential equation together with appropriate initial condition (Cauchy problem). Based on solution...

In this work we discuss higher order multi-term partial differential
equation (PDE) with the Caputo-Fabrizio fractional derivative in time. Using
method of separation of variables, we reduce fractional order partial differential
equation to the integer order. We represent explicit solution of formulated
problem in particular case by Fourier series.

In this work, we investigate an inverse source problem for multi-term fractional mixed type equation in a rectangular domain. We seek solutions in a form of series expansions using orthogonal basis obtained by using the method of a separation of variables. The obtained solutions involve multi-variable Mittag-Leffler functions, and hence, certain pr...

In this work, we investigate a unique solvability of a direct and inverse source problem for a time-fractional partial differential equation with the Caputo and Bessel operators. Using spectral expansion method, we give explicit forms of solutions to formulated problems in terms of multinomial Mittag-Leffler and first kind Bessel functions.

In the present paper, we discuss solvability questions of a non-local problem with integral form transmitting conditions for a mixed parabolic–hyperbolic type equation with the Caputo fractional derivative in a domain bounded by smooth curves. A uniqueness of the solution for a formulated problem we prove using energy integral method with some modi...

Initial value problem involving Atangana-Baleanu derivative is considered. An Explicit solution of the given problem is obtained by reducing the differential equation to Volterra integral equation of second kind and by using Laplace transform. To find the solution of the Volterra equation, the successive approximation method is used and a lemma sim...

We consider the unique solvability of an inverse-source problem with integral transmitting condition for a time-fractional mixed type equation in rectangular domain where the unknown source term depends only on the space variable. The solution is based on a series expansion using a bi-orthogonal basis in space, corresponding to a non-self-adjoint b...

In this work, we consider an initial problem for second order partial differential equations with Caputo fractional derivatives in the time-variable and Bessel operator in the space-variable. For non-local boundary conditions, we present a solution of this problem in an explicit form representing it by the Fourier-Bessel series. The obtained soluti...

In the present work, we have considered a non-local boundary problem with integral matching conditions for mixed type equation,involving fractional diffusion and wave equations. Using specific algorithm we find solution of considered problem in an explicit form. The proof is based on the method of characteristics, Green's function, Voltera integral...

In this work, we investigate a unique solvability of a direct and inverse source problem for a time-fractional partial differential equation with the Caputo and Bessel operators. Using spectral expansion method, we give explicit forms of solutions to formulated problems in terms of multinomial Mittag-Leffler and first kind Bessel functions.

In this work, we investigate a linear differential equation involving Caputo-Fabrizio fractional derivative of order $1<\beta\leq 2$. Under some assumptions the considered equation is reduced to an integer order differential equation and solutions for different cases are obtained in explicit forms. We also prove a uniqueness of a solution of an ini...

The main purpose of this paper is to study the existence of solutions for the following hybrid nonlinear fractional pantograph equation
$$
\left\{
\begin{aligned}
&D_{0+}^\alpha \left[\frac{x(t)}{f(t,x(t),x(\varphi(t)))}\right]=g(t,x(t),x(\rho(t))),\,\,0<t<1\\
&x(0)=0,
\end{aligned}
\right.
$$
where $\alpha\in (0,1)$, $\varphi$ and $\rho$ are funct...

In this work, we investigate a linear differential equation involving Caputo-Fabrizio fractional derivative of order $1<\beta\leq 2$. Under some assumptions the considered equation is reduced to an integer order differential equation and solutions for different cases are obtained in explicit forms. We also prove a uniqueness of a solution of an ini...

In this work, we consider a number of boundary-value problems for time-fractional heat equation with the recently introduced Caputo-Fabrizio derivative. Using the method of separation of variables, we prove a unique solvability of the stated problems. Moreover, we have found an explicit solution to certain initial value problem for Caputo-Fabrizio...

In the present work, we discuss a unique solvability of an inverse-source problem with integral transmitting condition for time-fractional mixed type equation in a rectangular domain, where the unknown source term depends on space variable only. The method of solution based on a series expansion using bi-orthogonal basis of space corresponding to a...

In this work, we investigate a linear differential equation involving Caputo-Fabrizio fractional derivative of order 1 < β ≤ 2. Under some assumptions the considered equation is reduced to an integer order differential equation and solutions for different cases are obtained in explicit forms. We also prove a uniqueness of a solution of an initial v...

In the present work, we investigate a uniqueness of solution of the inverse
source problem with non-local conditions for mixed parabolic-hyperbolic type
equation with Caputo fractional derivative. Solution of the problem we
represent as bi-orthogonal series with respect to space variable and will get
fractional order differential equations with res...

The main object of the present paper is to establish new fractional integral formulas (of Marichev-Saigo-Maeda type) involving the products of the multivariable H-functions and the first class of multivariable polynomials due to Srivastava and Garg. All the results derived here are of general character and can yield a number of (new and known) resu...

We aim at establishing certain new image formulas of generalized hypergeometric functions by applying the operators of fractional derivative involving Appell’s function F3(.) due to Saigo–Maeda. Furthermore, by employing some integral transforms on the resulting formulas, we presented some more image formulas. All the results derived here are of ge...

In this work, we investigate a boundary problem with non-local conditions for mixed parabolic–hyperbolic-type equation with three lines of type changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part, we use solution o...

In the paper, we investigate a local boundary value problem with transmitting condition of the integral form for mixed parabolic-hyperbolic equation with non-characteristic line of type changing. Theorem on strong solvability of the considered problem has been proved and integral representation of the solution is obtained in a functional space. Usi...

We aim at establishing certain new image formulas of generalized hypergeometric functions by applying the operators of fractional derivative involving Appell’s function F3(.) due to Saigo-Maeda. Furthermore, by employing some integral transforms on the resulting formulas, we presented some more image formulas. All the results derived here are of ge...

In the present paper we consider an inverse source problem for a time-fractional mixed parabolic-hyperbolic equation with Caputo derivatives. In the case when the hyperbolic part of the considered mixed-type equation is the wave equation, the uniqueness of the source and the solution are strongly influenced by the initial time and the problem is ge...

We investigate a boundary problem with integral gluing condition and nonlocal conditions, connecting various parts of the boundary of mixed domain for parabolic-hyperbolic equation with the Caputo fractional differential operator. The uniqueness of the solution is proved by the “abc” method. Applying method of Green’s function solution of the probl...

The main aim of the present work is an investigation of analogue of the Tricomi problem with integral sewing
condition for parabolic-hyperbolic equation with the fractional derivative. The uniqueness of the solution for considered problem we prove by the method of energy integrals. The existence of the solution have been proved by reducing the cons...

In this paper, we investigate a boundary problem with nonlocal conditions for mixed
parabolic–hyperbolic type equation with three lines of type changing. Considered domain
contains a rectangle as a parabolic part and three domains bounded by smooth curves
and type-changing lines as a hyperbolic part of the mixed domain. Applying method of
energy in...

Classifications --go to www.ams.org/msc to find your classifications: 35M12, 35P10 Abstract In the present paper we consider an inverse source problem for time-fractional mixed parabolic-hyperbolic equation with the Caputo derivative. In case, when hyperbolic part of the considered mixed type equation is wave equation, the uniqueness of source and...

In the present paper we consider an inverse source problem for
time-fractional mixed parabolic-hyperbolic equation with Caputo derivative.When
hyperbolic part of the considered mixed equation is wave type equation, the
uniqueness of source and solution are strongly influenced by initial time and
generally is ill-posed. However, when the hyperbolic...

In the present work we investigate the Tricomi problem with integral gluing condition for parabolic-hyperbolic equation with the Caputo fractional order derivative. Using the method of energy integrals we prove the uniqueness of the solution for considered problem. The existence will be proved using methods of ordinary differential equations, Fredh...

In the present work we investigate the Tricomi problem with integral gluing
condition for parabolic-hyperbolic equation with the Caputo fractional order
derivative. Using the method of energy integrals we prove the uniqueness of the
solution for considered problem. The existence will be proved using methods of
ordinary differential equations, Fredh...

In the present paper we study boundary problems with integral
gluing conditions for parabolic-hyperbolic type equation with
two lines of type changing involving the Caputo fractional
operator. The first problem is equivalently reduced to the system
of second kind Volterra integral equations, while the second
problem is reduced to the system of seco...

In this work, we study a boundary value problem with non-local conditions, by relating values of the unknown function with various characteristics. The parabolic-hyperbolic equation with three lines of type changing is equivalently reduced to a system of Volterra integral equations of the second kind.

In the present work we investigate a boundary problem with non-local conditions, connecting values of seeking function on various characteristics for parabolic-hyperbolic equation with three lines of type changing. The considered problem is equivalently reduced to the system of Volterra integral equations of the second kind.

In the present work we consider a boundary value problem with gluing
conditions of integral form for parabolic-hyperbolic type equation. We prove
that the considered problem has the Volterra property. The main tools used in
the work are related to the method of the integral equations and functional
analysis.

In this work, we deal with degenerate parabolic equations with three lines
of degeneration. Using ”a-b-c” method we prove the uniqueness theorems defining
conditions to parameters. We show nontrivial solutions for considered problems,
when uniqueness conditions to parameters, participating in the equations are not
fulfilled.

In the present work a non-local boundary value problem with special gluing conditions for mixed parabolic-hyperbolic equation with parameter is considered. The parabolic part of this equation is fractional analog of heat equation and the hyperbolic part is the telegraph equation. The considered problem is reduced, for positive values of the paramet...

In the present paper an analogue of the Holmgren problem for a three-dimensional elliptic equation with singular coefficients is studied for unique solvability. The uniqueness of the solution of the considered problem is proved by an energy integral method. Applying a method of Green’s function, the solution of the problem is found in an explicit f...

The present work devoted to the finding explicit solution of a boundary
problem with the Dirichlet-Neumann condition for elliptic equation with
singular coefficients in a quarter of ball. For this aim the method of Green's
function have been used. Since, found Green's function contains a
hypergeometric function of Appell, we had to deal with decomp...

In the present work, we investigate the Dirichlet problem for a three-dimensional (3D)
elliptic equation with two singular coefficients. We find four fundamental solutions of the
equation, containing hypergeometric functions of Appell. Then using an ‘‘a-b-c’’ method,
the uniqueness for the solution of the Dirichlet problem is proved. Applying a met...

The main result of the present work is the finding of fundamental solutions for a class of
three-dimensional singular elliptic equations with a parameter. The fundamental solutions
found contain Lauricella’s hypergeometric functions and, as particular cases, other special
functions such as Appel’s and Horn’s hypergeometric functions.

Analogs of the Tricomi and the Gellerstedt problems with integral gluing conditions for mixed parabolic-hyperbolic equation with parameter have been considered. The considered mixed-type equation consists of fractional diffusion and telegraph equation. The Tricomi problem is equivalently reduced to the second-kind Volterra integral equation, which...

In the present paper, a boundary-value problem with Neumann condition for a three-dimensional elliptic equation with singular coefficients is studied. The uniqueness theorem for the considered problem is proven by the energy integral method. A solution of the studied problem is found in an explicit form using the method of Green functions.

In 2002, J.M.Rassias (Uniqueness of quasi-regular solutions for bi-parabolic elliptic bi-hyperbolic Tricomi problem, Complex Variables, 47 (8) (2002), 707-718) imposed and investigated the bi-parabolic elliptic bi-hyperbolic mixed type partial differential equation of second order. In the present paper some boundary-value problems with non-local in...

The present work is devoted to the studying of a boundary-value problem with Neumann's
condition for three-dimensional elliptic equation with singular coefficients. The main result
is a proof of the unique solvability of the problem considered. An energy integral method
and a Green's function method were used as the main tools in the proof of the m...

We consider an equation $$ L_{\alpha ,\beta ,\gamma} (u) \equiv u_{xx} + u_{yy} + u_{zz} + \displaystyle \frac{{2\alpha}}{x}u_x + \displaystyle \frac{{2\beta}}{y}u_y + \displaystyle \frac{{2\gamma}}{z}u_z = 0 $$ in a domain ${\bf R}_3^ + \equiv {{({x,y,z}): x > 0, y > 0, z > 0}}$. Here $\alpha ,\beta ,\gamma$ are constants, moreover $0 < 2\alpha, 2...

In the present paper the unique solvability of two non-local problems for the mixed parabolic-hyperbolic type equation with complex spectral parameter is proved. Sectors for values of the spectral parameter where these problems have unique solutions are shown. Uniqueness of the solution is proved by the method of energy integral and existence is pr...

In the present work uniqueness of solution of boundary value problem with nonlocal
conditions for mixed parabolic type equation is proven by the method of
energy integrals. Moreover, using Fourier method eigenvalues and eigenfunctions
of the considered problem are found.

In the present paper we study the unique solvability of two non-local boundary value problems with continuous and special
gluing conditions for parabolic-hyperbolic type equations. The uniqueness of the solutions of the considered problems are
proven by the “abc” method. Existence theorems for the solutions of these problems are proven by the metho...

In the present paper two non-local problems with special gluing condition for parabolic-hyperbolic type equation with non-characteristic line of changing type with spectral parameter are considered. Values of required function at x=0, y>0 are directly connected with values of this function at x=1, y>0, and values of derivatives of the required func...

In this work two non-local problems for the parabolic-hyperbolic type equation with non-characteristic line of changing type
are considered. Unique solvability of these problems is proven. The uniqueness of the solution is proven by the method of
energy integrals and the existence is proven by the method of integral equations.

The unique solvability of the Tricomi problem for the parabolic–hyperbolic equation with
complex spectral parameter is proved. Uniqueness of the solution is shown by the method
of energy integral and existence by the method of integral equations.

## Questions

Question (1)

I need explicit form of $ln [E_{a,b}(z)]$.

## Projects

Projects (5)

The Methusalem initiative is a long-term programme from the Flemish government. It is a continuation of our Odysseus 1 Project on problems in analysis and partial differential equations. We are based at the Analysis & PDE Centre at Ghent University, Belgium.
For more details please see
https://analysis-pde.org/methusalem/

Bulletin of the Institute of Mathematics (BIM) was established by Institute of Mathematics after V.I. Romanovskiy Uzbekistan Academy of Sciences and Mathematical Society of Uzbekistan, and aimed to publish new results obtained in all branches of mathematics. Journal papers written in Uzbek, Russian or English languages will be considered for publication. Journal has been included to the list of scientific journals by Higher Attestation Committee in April, 2019.
Notice that, 6 issues per year of this online journal will be published and no limitation for number of pages of papers. If you wish to publish your research findings in BIM please go to page “For Authors”. Published papers can be downloaded from “Archive”.

In this project, we aim to investigate a different kind of nonlocal boundary value problems with integral conjugating conditions for mixed type equations consisting of the fractional diffusion equation and classical wave equations in composite domains.