Stochastic fractional evolution equations and their applications
Agamirza E. Bashirov, Arzu Ahmadova Partial observability of stochastic semilinear systems Journal Article Systems and Control Letters, 168 , pp. 105348, 2022. @article{Bashirov2022, title = {Partial observability of stochastic semilinear systems}, author = {Agamirza E. Bashirov, Arzu Ahmadova}, doi = {https://doi.org/10.1016/j.sysconle.2022.105348}, year = {2022}, date = {2022-10-01}, journal = {Systems and Control Letters}, volume = {168}, pages = {105348}, abstract = {In this article, we consider an Itô stochastic semilinear differential equation with unknown initial state and a linear observation system. It is proved that under a certain condition on the observability Gramian, the initial state of the equation can be recovered. This result is demonstrated by an example.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this article, we consider an Itô stochastic semilinear differential equation with unknown initial state and a linear observation system. It is proved that under a certain condition on the observability Gramian, the initial state of the equation can be recovered. This result is demonstrated by an example. |
Nazim I Mahmudov, Arzu Ahmadova Some Results on Backward Stochastic Differential Equations of Fractional Order Journal Article Qualitative Theory of Dynamical Systems, 21 (4), pp. 1-23, 2022. @article{Ahmadova2021bb, title = {Some Results on Backward Stochastic Differential Equations of Fractional Order}, author = {Nazim I Mahmudov, Arzu Ahmadova}, url = {https://doi.org/10.1007/s12346-022-00657-z}, doi = {https://doi.org/10.1007/s12346-022-00657-z}, year = {2022}, date = {2022-09-20}, journal = {Qualitative Theory of Dynamical Systems}, volume = {21}, number = {4}, pages = {1-23}, abstract = {In this article, we deal with fractional stochastic differential equations, so-called Caputo type fractional backward stochastic differential equations (Caputo fBSDEs, for short), and study the well-posedness of an adapted solution to Caputo fBSDEs of order $\alpha\in (1/2, 1)$ whose coefficients satisfy a Lipschitz condition. A novelty of the article is that we introduce a new weighted norm in the square integrable measurable function space that is useful for proving a fundamental lemma and its well-posedness. For this class of systems, we then show the coincidence between the notion of stochastic Volterra integral equation and the mild solution.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this article, we deal with fractional stochastic differential equations, so-called Caputo type fractional backward stochastic differential equations (Caputo fBSDEs, for short), and study the well-posedness of an adapted solution to Caputo fBSDEs of order $alphain (1/2, 1)$ whose coefficients satisfy a Lipschitz condition. A novelty of the article is that we introduce a new weighted norm in the square integrable measurable function space that is useful for proving a fundamental lemma and its well-posedness. For this class of systems, we then show the coincidence between the notion of stochastic Volterra integral equation and the mild solution. |
Arzu Ahmadova, Nazim Mahmudov I 2021. @online{Ahmadova2021bb, title = {Existence and uniqueness result of an adapted solution of a singular backward stochastic nonlinear Volterra integral equation}, author = {Arzu Ahmadova, Nazim I Mahmudov }, doi = {arXiv:2109.08950}, year = {2021}, date = {2021-05-30}, abstract = {Our main result in this paper is to establish a fundamental lemma to prove the global existence and uniqueness of an adapted solution to a singular backward stochastic nonlinear Volterra integral equation (for short singular BSVIE) of order alpha in ( 1/2 ; 1) under a weaker condition than Lipschitz one in Hilbert space.}, keywords = {}, pubstate = {published}, tppubtype = {online} } Our main result in this paper is to establish a fundamental lemma to prove the global existence and uniqueness of an adapted solution to a singular backward stochastic nonlinear Volterra integral equation (for short singular BSVIE) of order alpha in ( 1/2 ; 1) under a weaker condition than Lipschitz one in Hilbert space. |
Arzu Ahmadova, Nazim Mahmudov I Strong convergence of a Euler-Maruyama method for fractional stochastic Langevin equations Journal Article Mathematics and Computers in Simulation, 2021. @article{Ahmadova2021bb, title = {Strong convergence of a Euler-Maruyama method for fractional stochastic Langevin equations}, author = {Arzu Ahmadova, Nazim I Mahmudov }, doi = {10.1016/j.matcom.2021.05.037}, year = {2021}, date = {2021-05-10}, journal = {Mathematics and Computers in Simulation}, abstract = {The novelty of this paper is to derive a mild solution by means of recently defined Mittag-Leffler type functions of fractional stochastic Langevin equations of orders $ alpha in (1,2]$ and $beta in (0,1]$ whose coefficients satisfy standard Lipschitz and linear growth conditions. Then, we prove existence and uniqueness results of mild solution and show the coincidence between the notion of mild solution and integral equation. For this class of system, we construct fractional Euler-Maruyama method and establish new results on strong convergence of this method for fractional stochastic Langevin equations. We also introduce a general form of the nonlinear fractional stochastic Langevin equation and derive a general mild solution. Finally, the numerical examples are illustrated to verify the main theory.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The novelty of this paper is to derive a mild solution by means of recently defined Mittag-Leffler type functions of fractional stochastic Langevin equations of orders $ alpha in (1,2]$ and $beta in (0,1]$ whose coefficients satisfy standard Lipschitz and linear growth conditions. Then, we prove existence and uniqueness results of mild solution and show the coincidence between the notion of mild solution and integral equation. For this class of system, we construct fractional Euler-Maruyama method and establish new results on strong convergence of this method for fractional stochastic Langevin equations. We also introduce a general form of the nonlinear fractional stochastic Langevin equation and derive a general mild solution. Finally, the numerical examples are illustrated to verify the main theory. |
Arzu Ahmadova, Nazim Mahmudov I Asymptotic separation of solutions to fractional stochastic multi-term differential equations Journal Article Fractal and Fractional, 5 (4), pp. 256, 2021. @article{Ahmadova2021bb, title = {Asymptotic separation of solutions to fractional stochastic multi-term differential equations}, author = {Arzu Ahmadova, Nazim I Mahmudov}, doi = {https://doi.org/10.3390/fractalfract5040256}, year = {2021}, date = {2021-03-13}, journal = {Fractal and Fractional}, volume = {5}, number = {4}, pages = {256}, abstract = {In this paper, we study the exact asymptotic separation rate of two distinct solutions of Caputo stochastic multi-term differential equations (Caputo SMTDEs). Our goal in this paper is to establish results of the global existence and uniqueness and continuity dependence of the initial values of the solutions to Caputo SMTDEs with non-permutable matrices of order α∈(12,1) and β∈(0,1) whose coefficients satisfy a standard Lipschitz condition. For this class of systems, we then show the asymptotic separation property between two different solutions of Caputo SMTDEs with a more general condition based on λ. Furthermore, the asymptotic separation rate for the two distinct mild solutions reveals that our asymptotic results are general.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper, we study the exact asymptotic separation rate of two distinct solutions of Caputo stochastic multi-term differential equations (Caputo SMTDEs). Our goal in this paper is to establish results of the global existence and uniqueness and continuity dependence of the initial values of the solutions to Caputo SMTDEs with non-permutable matrices of order α∈(12,1) and β∈(0,1) whose coefficients satisfy a standard Lipschitz condition. For this class of systems, we then show the asymptotic separation property between two different solutions of Caputo SMTDEs with a more general condition based on λ. Furthermore, the asymptotic separation rate for the two distinct mild solutions reveals that our asymptotic results are general. |
Arzu Ahmadova, Nazim Mahmudov I Ulam-Hyers stability of Caputo type fractional stochastic neutral differential equations Journal Article STATISTICS & PROBABILITY LETTERS, 168 ( 108949), 2021, ISSN: 0167-7152. @article{Ahmadova2021, title = {Ulam-Hyers stability of Caputo type fractional stochastic neutral differential equations}, author = {Arzu Ahmadova, Nazim I. Mahmudov }, url = {https://www.sciencedirect.com/science/article/pii/S0167715220302522}, doi = { 10.1016/j.spl.2020.108949}, issn = { 0167-7152}, year = {2021}, date = {2021-01-01}, journal = {STATISTICS & PROBABILITY LETTERS}, volume = {168}, number = { 108949}, abstract = {The novelty of this research work is to establish stability results in Ulam-Hyers sense for the nonlinear fractional stochastic neutral differential equations system with the aid of weighted maximum norm and Ito's isometry in finite dimensional stochastic settings. }, keywords = {}, pubstate = {published}, tppubtype = {article} } The novelty of this research work is to establish stability results in Ulam-Hyers sense for the nonlinear fractional stochastic neutral differential equations system with the aid of weighted maximum norm and Ito's isometry in finite dimensional stochastic settings. |
Arzu Ahmadova, Nazim Mahmudov I Asymptotic stability analysis of Riemann-Liouville fractional stochastic neutral differential equations Journal Article Miskolc Mathematical Notes, (accepted), pp. 1-18, 2020, ISSN: 1787-2413. @article{Ahmadova2020bb, title = {Asymptotic stability analysis of Riemann-Liouville fractional stochastic neutral differential equations}, author = {Arzu Ahmadova, Nazim I. Mahmudov}, doi = {arXiv:2109.11493}, issn = {1787-2413}, year = {2020}, date = {2020-12-14}, journal = {Miskolc Mathematical Notes}, number = {accepted}, pages = {1-18}, abstract = {The novelty of our paper is to establish results on asymptotic stability of mild solutions in pth moment to Riemann-Liouville fractional stochastic neutral differential equations (for short Riemann-Liouville FSNDEs) using a Banach’s contraction mapping principle. The core point of this paper is to derive the mild solution of FSNDEs involving RiemannLiouville fractional time-derivative by applying the stochastic version of variation of constants formula. The results are obtained with the help of the theory of fractional differential equations, some properties of Mittag-Leffler functions and asymptotic analysis under the assumption that the corresponding fractional stochastic neutral dynamical system is asymptotically stable.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The novelty of our paper is to establish results on asymptotic stability of mild solutions in pth moment to Riemann-Liouville fractional stochastic neutral differential equations (for short Riemann-Liouville FSNDEs) using a Banach’s contraction mapping principle. The core point of this paper is to derive the mild solution of FSNDEs involving RiemannLiouville fractional time-derivative by applying the stochastic version of variation of constants formula. The results are obtained with the help of the theory of fractional differential equations, some properties of Mittag-Leffler functions and asymptotic analysis under the assumption that the corresponding fractional stochastic neutral dynamical system is asymptotically stable. |
Arzu Ahmadova, Nazim Mahmudov I Existence and uniqueness results for a class of fractional stochastic neutral differential equations Journal Article CHAOS SOLITONS & FRACTALS, 139 (110253), 2020, ISSN: 0960-0779. @article{Ahmadova2020, title = {Existence and uniqueness results for a class of fractional stochastic neutral differential equations}, author = {Arzu Ahmadova, Nazim I. Mahmudov}, url = {https://www.sciencedirect.com/science/article/pii/S0960077920306494}, doi = {10.1016/j.chaos.2020.110253}, issn = {0960-0779}, year = {2020}, date = {2020-10-01}, journal = {CHAOS SOLITONS & FRACTALS}, volume = {139}, number = {110253}, abstract = {In this paper, we investigate new results on the existence and uniqueness of mild solutions to stochastic neutral differential equations involving Caputo fractional time derivative operator with Lipschitz coefficients and under some Caratheodory-type conditions on the coefficients through the Picard approximation technique. To do so, we derive a stochastic version of variation of constants formula for Caputo fractional differential systems whose coefficients satisfy standard Lipschitz and non-Lipschitz conditions. The main points are to prove a coincidence between the integral equation and the mild solution by applying Ito's isometry, martingale representation theorem, and the weighted maximum norm for a class of fractional stochastic neutral differential equations. Finally, examples are provided to support the efficiency of the main theory. }, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper, we investigate new results on the existence and uniqueness of mild solutions to stochastic neutral differential equations involving Caputo fractional time derivative operator with Lipschitz coefficients and under some Caratheodory-type conditions on the coefficients through the Picard approximation technique. To do so, we derive a stochastic version of variation of constants formula for Caputo fractional differential systems whose coefficients satisfy standard Lipschitz and non-Lipschitz conditions. The main points are to prove a coincidence between the integral equation and the mild solution by applying Ito's isometry, martingale representation theorem, and the weighted maximum norm for a class of fractional stochastic neutral differential equations. Finally, examples are provided to support the efficiency of the main theory. |
Arzu Ahmadova Ismail T. Huseynov, Nazim Mahmudov I CONTROLLABILITY OF FRACTIONAL STOCHASTIC DELAY DYNAMICAL SYSTEMS Journal Article PROCEEDINGS OF THE INSTITUTE OF MATHEMATICS AND MECHANICS, 46 (2), pp. 294-320, 2020, ISSN: 2409-4986. @article{Ahmadova2020b, title = {CONTROLLABILITY OF FRACTIONAL STOCHASTIC DELAY DYNAMICAL SYSTEMS}, author = {Arzu Ahmadova, Ismail T. Huseynov, Nazim I. Mahmudov}, url = {http://proc.imm.az/volumes/46-2/46-02-10.pdf}, doi = {10.29228/proc.34}, issn = {2409-4986}, year = {2020}, date = {2020-09-01}, journal = {PROCEEDINGS OF THE INSTITUTE OF MATHEMATICS AND MECHANICS}, volume = {46}, number = {2}, pages = {294-320}, abstract = {In this paper, we consider Caputo type fractional stochastic time-delay system with permutable matrices. We derive stochastic analogue of variation of constants formula via a newly defined delayed Mittag-Leffler type matrix function. Thus, we investigate new results on existence and uniqueness of mild solutions with the help of weighted maximum norm to fractional stochastic time-delay differential equations whose coefficients satisfy standard Lipschitz conditions. The main points in the proof are to apply Ito's isometry and martingale representation theorem, and to point out the coincidence between the notion of the integral equation and the mild solution. Finally, we study complete controllability results for linear and nonlinear fractional stochastic delay dynamical systems with Wiener noise.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper, we consider Caputo type fractional stochastic time-delay system with permutable matrices. We derive stochastic analogue of variation of constants formula via a newly defined delayed Mittag-Leffler type matrix function. Thus, we investigate new results on existence and uniqueness of mild solutions with the help of weighted maximum norm to fractional stochastic time-delay differential equations whose coefficients satisfy standard Lipschitz conditions. The main points in the proof are to apply Ito's isometry and martingale representation theorem, and to point out the coincidence between the notion of the integral equation and the mild solution. Finally, we study complete controllability results for linear and nonlinear fractional stochastic delay dynamical systems with Wiener noise. |
Arzu Ahmadova, Nazim Mahmudov I A VARIATION OF CONSTANT FORMULA FOR FRACTIONAL STOCHASTIC NEUTRAL DIFFERENTIAL EQUATIONS Proceeding Minist Transport Commun & High Technologies Republic Azerbaijan; Baku State Univ, Inst Appl Math BAKU STATE UNIV, INST APPLIED MATHEMATICS, Z KHALILOV ST 23, BAKU, AZ 1148, AZERBAIJAN, 2 , 2020, ISBN: 978-9952-37-451-3. @proceedings{Ahmadova2020bb, title = {A VARIATION OF CONSTANT FORMULA FOR FRACTIONAL STOCHASTIC NEUTRAL DIFFERENTIAL EQUATIONS}, author = {Arzu Ahmadova, Nazim I. Mahmudov }, isbn = {978-9952-37-451-3}, year = {2020}, date = {2020-08-26}, journal = {PROCEEDINGS OF THE 7TH INTERNATIONAL CONFERENCE ON CONTROL AND OPTIMIZATION WITH INDUSTRIAL APPLICATIONS}, volume = {2}, pages = { 47-49}, publisher = {BAKU STATE UNIV, INST APPLIED MATHEMATICS}, address = {Z KHALILOV ST 23, BAKU, AZ 1148, AZERBAIJAN}, organization = {Minist Transport Commun & High Technologies Republic Azerbaijan; Baku State Univ, Inst Appl Math}, keywords = {}, pubstate = {published}, tppubtype = {proceedings} } |
Operator theory and Fractional evolution/differential equations and their applications
Ismail T Huseynov Arzu Ahmadova, ; Mahmudov, Nazim I Perturbation properties of fractional strongly continuous cosine and sine family operators Journal Article 30 (8), pp. 2911–2940., 2022. @article{Huseynov2021bb, title = {Perturbation properties of fractional strongly continuous cosine and sine family operators}, author = {Ismail T Huseynov, Arzu Ahmadova, and Nazim I Mahmudov}, doi = {http://www.aimspress.com/aimspress-data/era/2022/8/PDF/era-30-08-148.pdf}, year = {2022}, date = {2022-04-28}, volume = {30}, number = {8}, pages = {2911–2940.}, abstract = {Perturbation theory has long been a very useful tool in the hands of mathematicians and physicists. The purpose of this paper is to prove some perturbation results for infinitesimal generators of fractional strongly continuous cosine families. That is, we impose sufficient conditions such that A is the infinitesimal generator of a fractional strongly continuous cosine family in a Banach space X, and B is a bounded linear operator in X, then A + B is also the infinitesimal generator of a fractional strongly continuous cosine family in X. Our results coincide with the classical ones when α = 2. Furthermore, depending on commutativity condition of linear bounded operators, we propose the elegant closed-form formulas for uniformly continuous perturbed fractional operator cosine and sine functions. Finally, we present an example in the context of one-dimensional perturbed fractional wave equation to demonstrate the applicability of our theoretical results and we give some comparisons with the existing literature. }, keywords = {}, pubstate = {published}, tppubtype = {article} } Perturbation theory has long been a very useful tool in the hands of mathematicians and physicists. The purpose of this paper is to prove some perturbation results for infinitesimal generators of fractional strongly continuous cosine families. That is, we impose sufficient conditions such that A is the infinitesimal generator of a fractional strongly continuous cosine family in a Banach space X, and B is a bounded linear operator in X, then A + B is also the infinitesimal generator of a fractional strongly continuous cosine family in X. Our results coincide with the classical ones when α = 2. Furthermore, depending on commutativity condition of linear bounded operators, we propose the elegant closed-form formulas for uniformly continuous perturbed fractional operator cosine and sine functions. Finally, we present an example in the context of one-dimensional perturbed fractional wave equation to demonstrate the applicability of our theoretical results and we give some comparisons with the existing literature. |
Nazim I Mahmudov Arzu Ahmadova, Ismail Huseynov T A novel technique for solving Sobolev-type fractional multi-order evolution equations Journal Article Computational and Applied Mathematics, 41 (2), pp. 1-35, 2022. @article{Mahmudov2021b, title = {A novel technique for solving Sobolev-type fractional multi-order evolution equations}, author = {Nazim I Mahmudov, Arzu Ahmadova, Ismail T Huseynov}, doi = {https://doi.org/10.1007/s40314-022-01781-x}, year = {2022}, date = {2022-03-01}, journal = {Computational and Applied Mathematics}, volume = {41}, number = {2}, pages = {1-35}, abstract = {A strong inspiration for studying Sobolev-type fractional evolution equations comes from the fact that have been verified to be useful tools in the modeling of many physical processes. We introduce a novel technique for solving Sobolev-type fractional evolution equations with multi-orders in a Banach space. We propose a new Mittag-Leffler-type function which is generated by linear bounded operators and investigate their properties which are productive for checking the candidate solutions for multi-term fractional differential equations. Furthermore, we propose an exact analytical representation of solutions for multi-dimensional fractional-order dynamical systems with nonpermutable and permutable matrices.}, keywords = {}, pubstate = {published}, tppubtype = {article} } A strong inspiration for studying Sobolev-type fractional evolution equations comes from the fact that have been verified to be useful tools in the modeling of many physical processes. We introduce a novel technique for solving Sobolev-type fractional evolution equations with multi-orders in a Banach space. We propose a new Mittag-Leffler-type function which is generated by linear bounded operators and investigate their properties which are productive for checking the candidate solutions for multi-term fractional differential equations. Furthermore, we propose an exact analytical representation of solutions for multi-dimensional fractional-order dynamical systems with nonpermutable and permutable matrices. |
Arzu Ahmadova Ismail T. Huseynov, Nazim Mahmudov. I Existence and stability results on multi-dimensional fractional-order systems Journal Article The Rocky Mountain Journal of Mathematics, 52 (1), pp. 1-14, 2022. @article{Ahmadova2020bb, title = { Existence and stability results on multi-dimensional fractional-order systems}, author = {Arzu Ahmadova, Ismail T. Huseynov, Nazim I. Mahmudov.}, doi = {DOI: 10.1216/rmj.2022.52.1}, year = {2022}, date = {2022-02-01}, journal = {The Rocky Mountain Journal of Mathematics}, volume = {52}, number = {1}, pages = {1-14}, abstract = {The novelties of this paper is the study of existence and uniqueness results for a class of incommensurate fractional differential equation systems with general multiorders using the weighted infinity norm with respect to the classical Mittag-Leffler function via the contraction mapping principle. The lack of stability results in the Ulam–Hyers sense on fractional multidimensional differential equations motivates us to generalize our theory to stability analysis based on fixed point approach.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The novelties of this paper is the study of existence and uniqueness results for a class of incommensurate fractional differential equation systems with general multiorders using the weighted infinity norm with respect to the classical Mittag-Leffler function via the contraction mapping principle. The lack of stability results in the Ulam–Hyers sense on fractional multidimensional differential equations motivates us to generalize our theory to stability analysis based on fixed point approach. |
Ismail T Huseynov Arzu Ahmadova, ; Mahmudov, Nazim Perturbation theory for fractional evolution equations in a Banach space Online 2021. @online{Huseynov2021bb, title = {Perturbation theory for fractional evolution equations in a Banach space}, author = {Ismail T Huseynov, Arzu Ahmadova, and Nazim Mahmudov }, doi = {arXiv:2108.13188}, year = {2021}, date = {2021-08-30}, abstract = {A strong inspiration for studying perturbation theory for fractional evolution equations comes from the fact that they have proven to be useful tools in modeling many physical processes. In this paper, we study fractional evolution equations of order alpha in (1,2] associated with the infinitesimal generator of an operator fractional cosine function generated by bounded time-dependent perturbations in a Banach space. We show that the abstract fractional Cauchy problem associated with the infinitesimal generator A of a strongly continuous fractional cosine function remains uniformly well-posed under bounded time-dependent perturbation of A. We also provide some necessary special cases.}, keywords = {}, pubstate = {published}, tppubtype = {online} } A strong inspiration for studying perturbation theory for fractional evolution equations comes from the fact that they have proven to be useful tools in modeling many physical processes. In this paper, we study fractional evolution equations of order alpha in (1,2] associated with the infinitesimal generator of an operator fractional cosine function generated by bounded time-dependent perturbations in a Banach space. We show that the abstract fractional Cauchy problem associated with the infinitesimal generator A of a strongly continuous fractional cosine function remains uniformly well-posed under bounded time-dependent perturbation of A. We also provide some necessary special cases. |
Arzu Ahmadova Ismail T. Huseynov, Arran Fernandez Nazim Mahmudov I Trivariate Mittag-Leffler functions used to solve multi-order systems of fractional differential equations Journal Article COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, 97 (105735), 2021, ISSN: 1007-5704. @article{Ahmadova2021bb, title = {Trivariate Mittag-Leffler functions used to solve multi-order systems of fractional differential equations}, author = { Arzu Ahmadova, Ismail T. Huseynov, Arran Fernandez, Nazim I. Mahmudov}, url = {https://www.sciencedirect.com/science/article/pii/S1007570421000460}, doi = { 10.1016/j.cnsns.2021.105735}, issn = {1007-5704}, year = {2021}, date = {2021-06-01}, journal = {COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION}, volume = {97}, number = {105735}, abstract = {Linear systems of fractional differential equations have been studied from various points of view: applications to electric circuit theory, approximate solutions by numerical methods, and recently exact solutions by analytical methods. We discover here that, to obtain a fully closed-form solution in all cases, it is necessary to introduce a new type of Mittag-Leffler function involving triple series, and also to construct the associated fractional calculus operators, which we introduce and study in this paper. We then complete the rigorous analytical solutions for the aforesaid systems of fractional differential equations. As a consequence, comparing the solutions found here with the vector-matrix solutions known in the literature, we obtain explicit formulae for the elements of the 2 x 2 matrix Mittag-Leffler function. }, keywords = {}, pubstate = {published}, tppubtype = {article} } Linear systems of fractional differential equations have been studied from various points of view: applications to electric circuit theory, approximate solutions by numerical methods, and recently exact solutions by analytical methods. We discover here that, to obtain a fully closed-form solution in all cases, it is necessary to introduce a new type of Mittag-Leffler function involving triple series, and also to construct the associated fractional calculus operators, which we introduce and study in this paper. We then complete the rigorous analytical solutions for the aforesaid systems of fractional differential equations. As a consequence, comparing the solutions found here with the vector-matrix solutions known in the literature, we obtain explicit formulae for the elements of the 2 x 2 matrix Mittag-Leffler function. |
Arzu Ahmadova, Nazim Mahmudov I Langevin differential equations with general fractional orders and their applications to electric circuit theory Journal Article JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 388 (113299), 2021, ISSN: 0377-0427. @article{Ahmadova2021b, title = {Langevin differential equations with general fractional orders and their applications to electric circuit theory}, author = {Arzu Ahmadova, Nazim I. Mahmudov }, url = {https://www.sciencedirect.com/science/article/pii/S0377042720305902}, doi = {10.1016/j.cam.2020.113299}, issn = {0377-0427}, year = {2021}, date = {2021-05-01}, journal = {JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS}, volume = {388}, number = {113299}, abstract = {Multi-order fractional differential equations have been studied due to their applications in modeling, and solved using various mathematical methods. We present explicit analytical solutions for several families of Langevin differential equations with general fractional orders, both homogeneous and inhomogeneous cases. The results can be written, in general and special cases, by means of a recently defined bivariate Mittag-Leffler function and the associated operators of fractional calculus. The novelty of this work is to apply an appropriate norm on the proof of existence and uniqueness theorem, and discuss the application of Langevin differential equation with fractional orders in several interesting cases to the electrical circuit. Moreover, we investigate Ulam-Hyers stability of Caputo type fractional Langevin differential equation. At the end, we provide an example to verify our main results. }, keywords = {}, pubstate = {published}, tppubtype = {article} } Multi-order fractional differential equations have been studied due to their applications in modeling, and solved using various mathematical methods. We present explicit analytical solutions for several families of Langevin differential equations with general fractional orders, both homogeneous and inhomogeneous cases. The results can be written, in general and special cases, by means of a recently defined bivariate Mittag-Leffler function and the associated operators of fractional calculus. The novelty of this work is to apply an appropriate norm on the proof of existence and uniqueness theorem, and discuss the application of Langevin differential equation with fractional orders in several interesting cases to the electrical circuit. Moreover, we investigate Ulam-Hyers stability of Caputo type fractional Langevin differential equation. At the end, we provide an example to verify our main results. |
Ismail T Huseynov Arzu Ahmadova, Nazim Mahmudov On a study of Sobolev type fractional functional evolution equations Journal Article Mathematical Methods in the Applied Sciences, (accepted), 2021. @article{Huseynov2021b, title = {On a study of Sobolev type fractional functional evolution equations}, author = {Ismail T Huseynov, Arzu Ahmadova, Nazim Mahmudov}, url = {https://www.authorea.com/doi/full/10.22541/au.161562420.01059626}, doi = {10.22541/au.161562420.01059626/v1}, year = {2021}, date = {2021-03-13}, journal = {Mathematical Methods in the Applied Sciences}, number = {accepted}, abstract = {Sobolev type fractional functional evolution equations have many applications in the modeling of many physical processes. Therefore, we investigate fractional-order time-delay evolution equation of Sobolev type with multi-orders in a Banach space and introduce an analytical representation of a mild solution via a new delayed Mittag-Leffler type function which is generated by linear bounded operators. Furthermore, we derive an exact analytical representation of solutions for multi-dimensional fractional functional dynamical systems with nonpermutable and permutable matrices. We also study stability analysis of the given time-delay system in Ulam-Hyers sense with the help of Laplace transform}, keywords = {}, pubstate = {published}, tppubtype = {article} } Sobolev type fractional functional evolution equations have many applications in the modeling of many physical processes. Therefore, we investigate fractional-order time-delay evolution equation of Sobolev type with multi-orders in a Banach space and introduce an analytical representation of a mild solution via a new delayed Mittag-Leffler type function which is generated by linear bounded operators. Furthermore, we derive an exact analytical representation of solutions for multi-dimensional fractional functional dynamical systems with nonpermutable and permutable matrices. We also study stability analysis of the given time-delay system in Ulam-Hyers sense with the help of Laplace transform |
Ismail T. Huseynov Arzu Ahmadova, Arran Fernandez Nazim Mahmudov I Explicit analytical solutions of incommensurate fractional differential equation systems Journal Article Applied Mathematics and Computation, 390 (125590), 2021, ISSN: 0096-3003. @article{Huseynov2021, title = {Explicit analytical solutions of incommensurate fractional differential equation systems}, author = {Ismail T. Huseynov, Arzu Ahmadova, Arran Fernandez, Nazim I. Mahmudov}, url = {https://www.sciencedirect.com/science/article/pii/S0096300320305452}, doi = {10.1016/j.amc.2020.125590}, issn = {0096-3003}, year = {2021}, date = {2021-02-01}, journal = {Applied Mathematics and Computation}, volume = {390}, number = {125590}, abstract = {Fractional differential equations have been studied due to their applications in modelling, and solved using various mathematical methods. Systems of fractional differential equations are also used, for example in the study of electric circuits, but they are more difficult to analyse mathematically. We present explicit solutions for several families of such systems, both homogeneous and inhomogeneous cases, both commensurate and incommensurate. The results can be written, in several interesting special cases, in terms of a recently defined bivariate Mittag-Leffler function and the associated operators of fractional calculus}, keywords = {}, pubstate = {published}, tppubtype = {article} } Fractional differential equations have been studied due to their applications in modelling, and solved using various mathematical methods. Systems of fractional differential equations are also used, for example in the study of electric circuits, but they are more difficult to analyse mathematically. We present explicit solutions for several families of such systems, both homogeneous and inhomogeneous cases, both commensurate and incommensurate. The results can be written, in several interesting special cases, in terms of a recently defined bivariate Mittag-Leffler function and the associated operators of fractional calculus |
Ismail T. Huseynov Arzu Ahmadova, Nazim Mahmudov I 2020. @online{Huseynov2020b, title = {Fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives and their applications}, author = {Ismail T. Huseynov, Arzu Ahmadova, Nazim I. Mahmudov }, doi = {arXiv preprint arXiv:2012.11360}, year = {2020}, date = {2020-12-18}, abstract = {In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type differentiation operators with general fractional-order. A rule of fractional differentiation under integral sign with general order is necessary and applicable tool for verification by substitution for candidate solutions of inhomogeneous multi-term fractional differential equations. We derive explicit analytical solutions of generalized Bagley-Torvik equations in terms of recently defined bivariate Mittag-Leffler type functions that based on fractional Green's function method and verified solutions by substitution in accordance by applying the fractional Leibniz integral rule. Furthermore, we study an oscillator equation as a special case of differential equations with multi-orders via the Leibniz integral rule.}, keywords = {}, pubstate = {published}, tppubtype = {online} } In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type differentiation operators with general fractional-order. A rule of fractional differentiation under integral sign with general order is necessary and applicable tool for verification by substitution for candidate solutions of inhomogeneous multi-term fractional differential equations. We derive explicit analytical solutions of generalized Bagley-Torvik equations in terms of recently defined bivariate Mittag-Leffler type functions that based on fractional Green's function method and verified solutions by substitution in accordance by applying the fractional Leibniz integral rule. Furthermore, we study an oscillator equation as a special case of differential equations with multi-orders via the Leibniz integral rule. |
Arzu Ahmadova, Nazim Mahmudov I Application of Langevin equations with general fractional orders to electric circuit theory Proceeding 2020. @proceedings{Ahmadova2020bb, title = {Application of Langevin equations with general fractional orders to electric circuit theory}, author = {Arzu Ahmadova, Nazim I Mahmudov }, url = {https://www.researchgate.net/publication/346719358_Application_of_Langevin_equations_with_general_fractional_orders_to_electric_circuit_theory}, year = {2020}, date = {2020-12-06}, abstract = {Multi-order fractional differential equations have been studied due to their applications in modeling, and solved using various mathematical methods. We present explicit analytical solutions for several families of Langevin differential equations with general fractional orders, both homogeneous and inhomogeneous cases. The results can be written, in general and special cases, by means of a recently defined bivariate Mittag-Leffler function and the associated operators of fractional calculus. The novelty of this work is to apply an appropriate norm on the proof of existence and uniqueness theorem, and discuss the application of Langevin differential equation with fractional orders in several interesting case to the electrical circuit theory.}, keywords = {}, pubstate = {published}, tppubtype = {proceedings} } Multi-order fractional differential equations have been studied due to their applications in modeling, and solved using various mathematical methods. We present explicit analytical solutions for several families of Langevin differential equations with general fractional orders, both homogeneous and inhomogeneous cases. The results can be written, in general and special cases, by means of a recently defined bivariate Mittag-Leffler function and the associated operators of fractional calculus. The novelty of this work is to apply an appropriate norm on the proof of existence and uniqueness theorem, and discuss the application of Langevin differential equation with fractional orders in several interesting case to the electrical circuit theory. |
Ismail T Huseynov Arzu Ahmadova, Gbenga Ojo Nazim Mahmudov O I A natural extension of Mittag-Leffler function associated with a triple infinite series Online 2020. @online{Huseynov2020, title = {A natural extension of Mittag-Leffler function associated with a triple infinite series}, author = {Ismail T Huseynov, Arzu Ahmadova, Gbenga O Ojo, Nazim I Mahmudov,}, url = {https://arxiv.org/abs/2011.03999}, doi = {arXiv preprint arXiv:2011.03999}, year = {2020}, date = {2020-11-08}, abstract = {We establish a new natural extension of Mittag-Leffler function with three variables which is so called "trivariate Mittag-Leffler function". The trivariate Mittag-Leffler function can be expressed via complex integral representation by putting to use of the eminent Hankel's integral. We also investigate Laplace integral relation and convolution result for a univariate version of this function. Moreover, we present fractional derivative of trivariate Mittag-Leffler function in Caputo type and we also discuss Riemann- Liouville type fractional integral and derivative of this function. The link of trivariate Mittag-Leffler function with fractional differential equation systems involving different fractional orders is necessary on certain applications in physics. Thus, we provide an exact analytic solutions of homogeneous and inhomogeneous multi-term fractional differential equations by means of a newly defined trivariate Mittag-Leffler functions.}, keywords = {}, pubstate = {published}, tppubtype = {online} } We establish a new natural extension of Mittag-Leffler function with three variables which is so called "trivariate Mittag-Leffler function". The trivariate Mittag-Leffler function can be expressed via complex integral representation by putting to use of the eminent Hankel's integral. We also investigate Laplace integral relation and convolution result for a univariate version of this function. Moreover, we present fractional derivative of trivariate Mittag-Leffler function in Caputo type and we also discuss Riemann- Liouville type fractional integral and derivative of this function. The link of trivariate Mittag-Leffler function with fractional differential equation systems involving different fractional orders is necessary on certain applications in physics. Thus, we provide an exact analytic solutions of homogeneous and inhomogeneous multi-term fractional differential equations by means of a newly defined trivariate Mittag-Leffler functions. |