Emilia Bazhlekova - Academia.edu (original) (raw)

Papers by Emilia Bazhlekova

Research paper thumbnail of Subordination in a Class of Generalized Time-Fractional Diffusion-Wave Equations

Fractional Calculus and Applied Analysis, 2018

Motivated by recently proposed generalizations of the diffusion-wave equation with the Caputo tim... more Motivated by recently proposed generalizations of the diffusion-wave equation with the Caputo time fractional derivative of order α ∈ (1, 2), in the present survey paper a class of generalized time-fractional diffusion-wave equations is introduced. Its definition is based on the subordination principle for Volterra integral equations and involves the notion of complete Bernstein function. Various members of this class are surveyed, including the distributed-order time-fractional diffusion-wave equation and equations governing wave propagation in viscoelastic media with completely monotone relaxation moduli.

Research paper thumbnail of Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model

Fractal and fractional, Aug 19, 2023

This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Research paper thumbnail of Existence and uniqueness results for a fractional evolution equation in Hilbert space

Fractional Calculus and Applied Analysis, Mar 18, 2012

The existence and uniqueness of the solution of a fractional evolution equation with the Riemann-... more The existence and uniqueness of the solution of a fractional evolution equation with the Riemann-Liouville fractional derivative of order α ∈ (0, 1) is studied in Hilbert space, based on the theory of sums of accretive operators. The results are applied to some subdiffusion problems.

Research paper thumbnail of Alternating direction implicit schemes for two-dimensional generalized fractional Oldroyd-B fluids

Nucleation and Atmospheric Aerosols, 2015

The two-dimensional Rayleigh-Stokes problem for a generalized fractional Oldroyd-B fluid is consi... more The two-dimensional Rayleigh-Stokes problem for a generalized fractional Oldroyd-B fluid is considered in the present work. First and second order approximations of the fractional time derivatives are implemented in the developed alternating direction implicit finite difference schemes. Second and compact fourth order approximations are used for the space derivatives. Extensive numerical experiments are performed in order to investigate the stability and accuracy of the proposed algorithms.

Research paper thumbnail of Identification of a space-dependent source term in a nonlocal problem for the general time-fractional diffusion equation

Journal of Computational and Applied Mathematics, Apr 1, 2021

This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Research paper thumbnail of Alternating direction implicit schemes for two-dimensional generalized fractional Oldroyd-B fluids

AIP Conference Proceedings, 2015

The two-dimensional Rayleigh-Stokes problem for a generalized fractional Oldroyd-B fluid is consi... more The two-dimensional Rayleigh-Stokes problem for a generalized fractional Oldroyd-B fluid is considered in the present work. First and second order approximations of the fractional time derivatives are implemented in the developed alternating direction implicit finite difference schemes. Second and compact fourth order approximations are used for the space derivatives. Extensive numerical experiments are performed in order to investigate the stability and accuracy of the proposed algorithms.

Research paper thumbnail of RESEARCH ARTICLE Exact solution of two-term time-fractional Thornley's problem by operational method

Research paper thumbnail of Subordination Principle for Generalized Fractional Zener Models

Fractal and Fractional

The fractional Zener constitutive law is frequently used as a model of solid-like viscoelastic be... more The fractional Zener constitutive law is frequently used as a model of solid-like viscoelastic behavior. In this work, a class of linear viscoelastic models of Zener type, which generalize the fractional Zener model, is studied by the use of Bernstein functions technique. We prove that the corresponding relaxation moduli are completely monotone functions under appropriate thermodynamic restrictions on the parameters. Based on this property, we study the propagation function and establish the subordination principle for the corresponding Zener-type wave equation, which provides an integral representation of the solution in terms of the propagation function and the solution of a related classical wave equation. The analytical findings are supported by numerical examples.

Research paper thumbnail of Strict L p Solutions for Nonautonomous

Research paper thumbnail of Fractional derivative modeling of bioreaction-diffusion processes

THERMOPHYSICAL BASIS OF ENERGY TECHNOLOGIES (TBET 2020), 2021

The aim of this work is to analyze the application of fractional derivatives in time for the math... more The aim of this work is to analyze the application of fractional derivatives in time for the mathematical modeling of complex processes. As an example, a bioprocess is considered and the related distribution of nutrients, bacteria and bioproduct in multiphase fluid systems. The mathematical model under investigation includes convection, diffusion, bioreaction and boundary conditions at the interfaces: interface mass-transfer and adsorption. Different approaches of time-fractional modeling of a bioprocess are discussed. To evaluate the ability of a fractional order model to correctly reproduce the behavior that the underlying process must exhibit, numerical procedures are developed and computer simulations are performed.

Research paper thumbnail of On the Rayleigh-Stokes problem for generalized fractional Oldroyd-B fluids

AIP Conference Proceedings, 2015

We consider the initial-boundary value problem for the velocity distribution of a unidirectional ... more We consider the initial-boundary value problem for the velocity distribution of a unidirectional flow of a generalized Oldroyd-B fluid with fractional derivative model. It involves two different Riemann-Liouville fractional derivatives in time. The problem is studied in a general abstract setting, based on a reformulation as a Volterra integral equation with kernel represented in terms of Mittag-Leffler functions. Special attention is paid to the solution behavior in the scalar case, using some facts of the theory of the Bernstein functions. Numerical experiments are performed for different values of the parameters and plots are presented and discussed. The results are compared to those obtained in the limiting cases of generalized fractional Maxwell and second grade fluids.

Research paper thumbnail of Wave Propagation in Viscoelastic Half-Space with Memory Functions of Mittag-Leffler Type

International Journal of Applied Mathematics, Jun 30, 2021

The problem of one-dimensional non-stationary wave propagation in viscoelastic half space is stud... more The problem of one-dimensional non-stationary wave propagation in viscoelastic half space is studied. For the description of the hereditary properties of the viscoelastic medium, several examples of completely monotone relaxation kernels are considered. They are expressed in terms of functions of Mittag-Leffler type, including the recently introduced multinomial Prabhakar type function. Applying Laplace transform in time, some characteristics of the propagation function are discussed, such as non-negativity, monotonicity, propagation speed, presence/absence of wave front, and explicit integral representation of the solution is derived.

Research paper thumbnail of Application of Dimovski ’ S Convolutional Calculus to Distributed-Order Time-Fractional Diffusion Equation on a Bounded Domain

An initial-boundary value problem for the one-dimensional timefractional diffusion equation of di... more An initial-boundary value problem for the one-dimensional timefractional diffusion equation of distributed order is considered. Applying the convolutional calculus approach proposed by Dimovski (I.H. Dimovski, Convolutional Calculus, Kluwer, Dordrecht (1990)), a Duhamel-type representation of the solution is found in the form of a convolution product of a particular solution and the given initial function. A non-classical convolution with respect to the spatial variable is used. The particular solution is found by eigenfunction expansion. Special attention is paid to the study of the time-dependent components in this expansion. It is proven that the obtained solution is a solution in the classical sense. The Duhamel-type representation is used for numerical computation of the solution in some numerical examples.

Research paper thumbnail of Subordination approach to multi-term time-fractional diffusion-wave equations

This paper is concerned with the fractional evolution equation with a discrete distribution of Ca... more This paper is concerned with the fractional evolution equation with a discrete distribution of Caputo time-derivatives such that the largest and the smallest orders, α and α_m, satisfy the conditions 1<α< 2 and α-α_m< 1. First, based on a study of the related propagation function, the nonnegativity of the fundamental solutions to the spatially one-dimensional Cauchy and signaling problems is proven and propagation speed of a disturbance is discussed. Next, we study the equation with a general linear spatial differential operator defined in a Banach space and suppose it generates a cosine family. A subordination principle is established, which implies the existence of a unique solution and gives an integral representation of the solution operator in terms of the corresponding cosine family and a probability density function. Explicit representation of the probability density function is derived. The subordination principle is applied for obtaining regularity results. The ana...

Research paper thumbnail of An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a... more An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.

Research paper thumbnail of Strict LpSolutions for Nonautonomous Fractional Evolution Equations

Nonautonomous problems are important especially as a transient case between the linear and the no... more Nonautonomous problems are important especially as a transient case between the linear and the nonlinear theory. We study the nonautonomous linear problem for the fractional evolution equation D t u(t) + A(t)u(t) = f(t), a.a. t ∈ (0, T ), where D t is the Riemann-Liouville fractional derivative of order α ∈ (0, 1), {A(t)}t∈[0,T ] is a family of linear closed operators densely defined on a Banach space X and the forcing function f(t) ∈ L(0, T ; X). Strict L solvability of this problem is proved for a suitable class of operators A(t). The proof is based on L regularity estimates for the corresponding autonomous problem.

Research paper thumbnail of Completely monotone multinomial mittag-leffler type functions and diffusion equations with multiple time-derivatives

Fractional Calculus and Applied Analysis

The multinomial Mittag-Leffler function plays a crucial role in the study of multi-term time-frac... more The multinomial Mittag-Leffler function plays a crucial role in the study of multi-term time-fractional evolution equations. In this work we establish basic properties of the Prabhakar type generalization of this function with the main emphasis on complete monotonicity. As particular examples, the relaxation functions for equations with multiple time-derivatives in the so-called “natural” and “modified” forms are studied in detail and useful estimates are derived. The obtained results extend known properties of the classical Mittag-Leffler function. The main tools used in this work are Laplace transform and Bernstein functions’ technique.

Research paper thumbnail of Fundamental solution of a three-dimensional fractional Jeffreys-type heat equation

THERMOPHYSICAL BASIS OF ENERGY TECHNOLOGIES (TBET 2020)

The three-dimensional Cauchy problem for the heat conduction equation with a fractional Jeffreys-... more The three-dimensional Cauchy problem for the heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Two different cases are distinguished: diffusion and propagation regimes. In the diffusion regime the three-dimensional fundamental solution is shown to be a spatial probability density function evolving in time. In the propagation regime the solution can have negative values, and therefore, does not allow a probabilistic interpretation. Explicit integral representation for the threedimensional fundamental solution is derived and used for numerical experiments.

Research paper thumbnail of Identification of a space-dependent source term in a nonlocal problem for the general time-fractional diffusion equation

Journal of Computational and Applied Mathematics

Research paper thumbnail of Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation

Fractal and Fractional

The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Dependi... more The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized diffusion equation, while in the second it is a generalized wave equation. The corresponding memory kernels are expressed in both cases in terms of Mittag–Leffler functions. Explicit representations for the one-dimensional fundamental solution and the mean squared displacement are provided and analyzed analytically and numerically. The one-dimensional fundamental solution is shown to be a spatial probability density function evolving in time, which is unimodal in the diffusion regime and bimodal in the propagation regime. The multi-dimensional fundamental solutions are probability densities only in the diffusion case, while in the propagation case they can have negative values. In...

Research paper thumbnail of Subordination in a Class of Generalized Time-Fractional Diffusion-Wave Equations

Fractional Calculus and Applied Analysis, 2018

Motivated by recently proposed generalizations of the diffusion-wave equation with the Caputo tim... more Motivated by recently proposed generalizations of the diffusion-wave equation with the Caputo time fractional derivative of order α ∈ (1, 2), in the present survey paper a class of generalized time-fractional diffusion-wave equations is introduced. Its definition is based on the subordination principle for Volterra integral equations and involves the notion of complete Bernstein function. Various members of this class are surveyed, including the distributed-order time-fractional diffusion-wave equation and equations governing wave propagation in viscoelastic media with completely monotone relaxation moduli.

Research paper thumbnail of Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model

Fractal and fractional, Aug 19, 2023

This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Research paper thumbnail of Existence and uniqueness results for a fractional evolution equation in Hilbert space

Fractional Calculus and Applied Analysis, Mar 18, 2012

The existence and uniqueness of the solution of a fractional evolution equation with the Riemann-... more The existence and uniqueness of the solution of a fractional evolution equation with the Riemann-Liouville fractional derivative of order α ∈ (0, 1) is studied in Hilbert space, based on the theory of sums of accretive operators. The results are applied to some subdiffusion problems.

Research paper thumbnail of Alternating direction implicit schemes for two-dimensional generalized fractional Oldroyd-B fluids

Nucleation and Atmospheric Aerosols, 2015

The two-dimensional Rayleigh-Stokes problem for a generalized fractional Oldroyd-B fluid is consi... more The two-dimensional Rayleigh-Stokes problem for a generalized fractional Oldroyd-B fluid is considered in the present work. First and second order approximations of the fractional time derivatives are implemented in the developed alternating direction implicit finite difference schemes. Second and compact fourth order approximations are used for the space derivatives. Extensive numerical experiments are performed in order to investigate the stability and accuracy of the proposed algorithms.

Research paper thumbnail of Identification of a space-dependent source term in a nonlocal problem for the general time-fractional diffusion equation

Journal of Computational and Applied Mathematics, Apr 1, 2021

This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Research paper thumbnail of Alternating direction implicit schemes for two-dimensional generalized fractional Oldroyd-B fluids

AIP Conference Proceedings, 2015

The two-dimensional Rayleigh-Stokes problem for a generalized fractional Oldroyd-B fluid is consi... more The two-dimensional Rayleigh-Stokes problem for a generalized fractional Oldroyd-B fluid is considered in the present work. First and second order approximations of the fractional time derivatives are implemented in the developed alternating direction implicit finite difference schemes. Second and compact fourth order approximations are used for the space derivatives. Extensive numerical experiments are performed in order to investigate the stability and accuracy of the proposed algorithms.

Research paper thumbnail of RESEARCH ARTICLE Exact solution of two-term time-fractional Thornley's problem by operational method

Research paper thumbnail of Subordination Principle for Generalized Fractional Zener Models

Fractal and Fractional

The fractional Zener constitutive law is frequently used as a model of solid-like viscoelastic be... more The fractional Zener constitutive law is frequently used as a model of solid-like viscoelastic behavior. In this work, a class of linear viscoelastic models of Zener type, which generalize the fractional Zener model, is studied by the use of Bernstein functions technique. We prove that the corresponding relaxation moduli are completely monotone functions under appropriate thermodynamic restrictions on the parameters. Based on this property, we study the propagation function and establish the subordination principle for the corresponding Zener-type wave equation, which provides an integral representation of the solution in terms of the propagation function and the solution of a related classical wave equation. The analytical findings are supported by numerical examples.

Research paper thumbnail of Strict L p Solutions for Nonautonomous

Research paper thumbnail of Fractional derivative modeling of bioreaction-diffusion processes

THERMOPHYSICAL BASIS OF ENERGY TECHNOLOGIES (TBET 2020), 2021

The aim of this work is to analyze the application of fractional derivatives in time for the math... more The aim of this work is to analyze the application of fractional derivatives in time for the mathematical modeling of complex processes. As an example, a bioprocess is considered and the related distribution of nutrients, bacteria and bioproduct in multiphase fluid systems. The mathematical model under investigation includes convection, diffusion, bioreaction and boundary conditions at the interfaces: interface mass-transfer and adsorption. Different approaches of time-fractional modeling of a bioprocess are discussed. To evaluate the ability of a fractional order model to correctly reproduce the behavior that the underlying process must exhibit, numerical procedures are developed and computer simulations are performed.

Research paper thumbnail of On the Rayleigh-Stokes problem for generalized fractional Oldroyd-B fluids

AIP Conference Proceedings, 2015

We consider the initial-boundary value problem for the velocity distribution of a unidirectional ... more We consider the initial-boundary value problem for the velocity distribution of a unidirectional flow of a generalized Oldroyd-B fluid with fractional derivative model. It involves two different Riemann-Liouville fractional derivatives in time. The problem is studied in a general abstract setting, based on a reformulation as a Volterra integral equation with kernel represented in terms of Mittag-Leffler functions. Special attention is paid to the solution behavior in the scalar case, using some facts of the theory of the Bernstein functions. Numerical experiments are performed for different values of the parameters and plots are presented and discussed. The results are compared to those obtained in the limiting cases of generalized fractional Maxwell and second grade fluids.

Research paper thumbnail of Wave Propagation in Viscoelastic Half-Space with Memory Functions of Mittag-Leffler Type

International Journal of Applied Mathematics, Jun 30, 2021

The problem of one-dimensional non-stationary wave propagation in viscoelastic half space is stud... more The problem of one-dimensional non-stationary wave propagation in viscoelastic half space is studied. For the description of the hereditary properties of the viscoelastic medium, several examples of completely monotone relaxation kernels are considered. They are expressed in terms of functions of Mittag-Leffler type, including the recently introduced multinomial Prabhakar type function. Applying Laplace transform in time, some characteristics of the propagation function are discussed, such as non-negativity, monotonicity, propagation speed, presence/absence of wave front, and explicit integral representation of the solution is derived.

Research paper thumbnail of Application of Dimovski ’ S Convolutional Calculus to Distributed-Order Time-Fractional Diffusion Equation on a Bounded Domain

An initial-boundary value problem for the one-dimensional timefractional diffusion equation of di... more An initial-boundary value problem for the one-dimensional timefractional diffusion equation of distributed order is considered. Applying the convolutional calculus approach proposed by Dimovski (I.H. Dimovski, Convolutional Calculus, Kluwer, Dordrecht (1990)), a Duhamel-type representation of the solution is found in the form of a convolution product of a particular solution and the given initial function. A non-classical convolution with respect to the spatial variable is used. The particular solution is found by eigenfunction expansion. Special attention is paid to the study of the time-dependent components in this expansion. It is proven that the obtained solution is a solution in the classical sense. The Duhamel-type representation is used for numerical computation of the solution in some numerical examples.

Research paper thumbnail of Subordination approach to multi-term time-fractional diffusion-wave equations

This paper is concerned with the fractional evolution equation with a discrete distribution of Ca... more This paper is concerned with the fractional evolution equation with a discrete distribution of Caputo time-derivatives such that the largest and the smallest orders, α and α_m, satisfy the conditions 1<α< 2 and α-α_m< 1. First, based on a study of the related propagation function, the nonnegativity of the fundamental solutions to the spatially one-dimensional Cauchy and signaling problems is proven and propagation speed of a disturbance is discussed. Next, we study the equation with a general linear spatial differential operator defined in a Banach space and suppose it generates a cosine family. A subordination principle is established, which implies the existence of a unique solution and gives an integral representation of the solution operator in terms of the corresponding cosine family and a probability density function. Explicit representation of the probability density function is derived. The subordination principle is applied for obtaining regularity results. The ana...

Research paper thumbnail of An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a... more An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.

Research paper thumbnail of Strict LpSolutions for Nonautonomous Fractional Evolution Equations

Nonautonomous problems are important especially as a transient case between the linear and the no... more Nonautonomous problems are important especially as a transient case between the linear and the nonlinear theory. We study the nonautonomous linear problem for the fractional evolution equation D t u(t) + A(t)u(t) = f(t), a.a. t ∈ (0, T ), where D t is the Riemann-Liouville fractional derivative of order α ∈ (0, 1), {A(t)}t∈[0,T ] is a family of linear closed operators densely defined on a Banach space X and the forcing function f(t) ∈ L(0, T ; X). Strict L solvability of this problem is proved for a suitable class of operators A(t). The proof is based on L regularity estimates for the corresponding autonomous problem.

Research paper thumbnail of Completely monotone multinomial mittag-leffler type functions and diffusion equations with multiple time-derivatives

Fractional Calculus and Applied Analysis

The multinomial Mittag-Leffler function plays a crucial role in the study of multi-term time-frac... more The multinomial Mittag-Leffler function plays a crucial role in the study of multi-term time-fractional evolution equations. In this work we establish basic properties of the Prabhakar type generalization of this function with the main emphasis on complete monotonicity. As particular examples, the relaxation functions for equations with multiple time-derivatives in the so-called “natural” and “modified” forms are studied in detail and useful estimates are derived. The obtained results extend known properties of the classical Mittag-Leffler function. The main tools used in this work are Laplace transform and Bernstein functions’ technique.

Research paper thumbnail of Fundamental solution of a three-dimensional fractional Jeffreys-type heat equation

THERMOPHYSICAL BASIS OF ENERGY TECHNOLOGIES (TBET 2020)

The three-dimensional Cauchy problem for the heat conduction equation with a fractional Jeffreys-... more The three-dimensional Cauchy problem for the heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Two different cases are distinguished: diffusion and propagation regimes. In the diffusion regime the three-dimensional fundamental solution is shown to be a spatial probability density function evolving in time. In the propagation regime the solution can have negative values, and therefore, does not allow a probabilistic interpretation. Explicit integral representation for the threedimensional fundamental solution is derived and used for numerical experiments.

Research paper thumbnail of Identification of a space-dependent source term in a nonlocal problem for the general time-fractional diffusion equation

Journal of Computational and Applied Mathematics

Research paper thumbnail of Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation

Fractal and Fractional

The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Dependi... more The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized diffusion equation, while in the second it is a generalized wave equation. The corresponding memory kernels are expressed in both cases in terms of Mittag–Leffler functions. Explicit representations for the one-dimensional fundamental solution and the mean squared displacement are provided and analyzed analytically and numerically. The one-dimensional fundamental solution is shown to be a spatial probability density function evolving in time, which is unimodal in the diffusion regime and bimodal in the propagation regime. The multi-dimensional fundamental solutions are probability densities only in the diffusion case, while in the propagation case they can have negative values. In...