On a Boundary-Initial Value Problem for Fractional Differential Equation with Sequential Caputo derivatives

Fayziev Yusuf; Jumaeva Shakhnoza

arxiv: 2604.21319 · v1 · submitted 2026-04-23 · 🧮 math.NA · cs.NA· math.AP

On a Boundary-Initial Value Problem for Fractional Differential Equation with Sequential Caputo derivatives

Fayziev Yusuf , Jumaeva Shakhnoza This is my paper

Pith reviewed 2026-05-09 21:34 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.AP

keywords fractional differential equationssequential Caputo derivativesbivariate Mittag-Leffler functionanalytic solutionboundary-initial value problemL1-finite element methodnumerical scheme

0 comments

The pith

The boundary-initial value problem for a fractional differential equation with sequential Caputo derivatives admits an exact solution in terms of the bivariate Mittag-Leffler function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines a fractional differential equation that incorporates sequential Caputo derivatives to capture multiple memory effects in a system. The authors derive a closed-form analytic solution for the associated boundary-initial value problem by applying solution techniques suited to sequential fractional operators. They also establish supporting properties of the bivariate Mittag-Leffler function and construct a numerical approximation scheme that reformulates the problem sequentially before applying an L1-finite element discretization. A reader would care because explicit solutions remain uncommon in fractional models, and having both an exact formula and a matching numerical method allows direct verification and deeper insight into how memory accumulates across derivative orders.

Core claim

The authors establish that the solution of the boundary-initial value problem can be expressed exactly using the bivariate Mittag-Leffler function after applying standard operational properties of sequential Caputo derivatives; they further show that this representation holds under the stated initial and boundary data without requiring extra compatibility conditions, and they supply a set of auxiliary identities for the bivariate Mittag-Leffler function that facilitate the derivation.

What carries the argument

The bivariate Mittag-Leffler function, which encodes the combined action of the two sequential Caputo operators and directly supplies the closed-form solution coefficients.

If this is right

The exact formula supplies a benchmark against which any numerical method for sequential Caputo problems can be tested.
The listed properties of the bivariate Mittag-Leffler function become reusable tools for constructing solutions to other linear sequential fractional equations.
The sequential reformulation used for the numerical scheme converts the original problem into a system that existing L1-finite element codes can handle with only minor adjustments.
Analytic expressions of this type allow direct study of long-time decay rates and stability without discretizing time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same solution strategy could be tested on equations that mix sequential Caputo derivatives with Riemann-Liouville operators to see whether the bivariate Mittag-Leffler form survives.
The numerical scheme might be extended to nonlinear problems by treating the nonlinearity explicitly while retaining the linear fractional part in closed form.
Because the bivariate Mittag-Leffler function reduces to ordinary Mittag-Leffler functions when the two orders coincide, the present result specializes to known single-order cases and thereby recovers earlier literature as a consistency check.

Load-bearing premise

The sequential Caputo operators together with the given boundary-initial conditions permit an immediate representation by the bivariate Mittag-Leffler function without extra regularity or compatibility requirements on the data.

What would settle it

A specific choice of initial data and forcing term for which the explicit Mittag-Leffler series fails to satisfy the differential equation or the boundary conditions after direct substitution and differentiation.

Figures

Figures reproduced from arXiv: 2604.21319 by Fayziev Yusuf, Jumaeva Shakhnoza.

**Figure 1.** Figure 1: shows our numerical result (uh) compared side-by-side with the exact analytical solution (uex). The two curves overlap almost perfectly across the entire time range. This overlap confirms that our software and the chosen calculation method are highly reliable, even during the very beginning of the process [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗

**Figure 2.** Figure 2: Pointwise traces uh(0.5, t) in the admissible region. (A) Effect of α on the solution. (B) Effect of β on the solution. When we analyzed the total energy of the system ( [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: Energy decay and the ”rebound” effect over time. (A) Energy dependence ( L 2 norm) on α. (B) H1 norm for varying α [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

**Figure 4.** Figure 4: Combined influence of α and β over the admissible wedge. (A) Heatmap of numerical results. (B) Iso-response curves for parameters. 6. Conclusion In this study, we investigated a boundary–initial value problem for a fractional differential equation with sequential Caputo derivatives. Using the Fourier method, we proved the existence and uniqueness of a regular solution. We established the system’s well-po… view at source ↗

read the original abstract

In this paper, we investigate a fractional differential equation involving sequential Caputo derivatives, motivated by recent research on fractional models with multiple memory effects. Using techniques inspired by earlier works on sequential fractional operators, we derive the exact analytic solution of the problem in terms of the bivariate Mittag-Leffler function. Additionally, several useful properties of the bivariate Mittag-Leffler function are formulated to support the solution construction. Furthermore, we develop a numerical scheme using a sequential reformulation and the L1-finite element method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an exact bivariate Mittag-Leffler solution for a sequential Caputo boundary-initial problem plus an L1-FEM scheme, but the exact claim probably needs unstated compatibility conditions on the data.

read the letter

The main point with this paper is that the authors derive an exact solution expressed with the bivariate Mittag-Leffler function for their fractional differential equation that uses sequential Caputo derivatives, and they back it up with a numerical method based on L1 finite elements. They do well by pulling together the techniques from earlier sequential operator papers and applying them directly to this boundary-initial setup. Formulating the properties of the bivariate function is a helpful step that makes the solution construction clearer. The numerical scheme is presented as a way to handle cases where the analytic form isn't enough, which is a reasonable addition for a numerical analysis paper. Where it might be soft is on the conditions for that exact solution. The stress-test raises a valid concern: getting precisely the bivariate ML term without remainders usually requires the initial and boundary data to be compatible in a fractional sense. Nothing in the abstract indicates they impose or verify those relations, so the result could be conditional on data that satisfy them. If the full derivation doesn't address this, it would be good to see that clarified. The soundness looks okay on the surface since it builds on established Mittag-Leffler properties, but without the full text it's hard to confirm the rigor of the boundary condition incorporation. The work is solid enough on its own terms for what it is. This is aimed at researchers focused on fractional calculus applications with multiple derivatives, particularly those looking for benchmark solutions or numerical approaches in that area. A reader who needs explicit forms for such problems would find it useful. I would recommend sending it for peer review. It's a focused contribution that deserves referee feedback, especially to check the compatibility aspect and the numerical convergence details.

Referee Report

2 major / 1 minor

Summary. The paper investigates a boundary-initial value problem for a fractional differential equation involving sequential Caputo derivatives. It derives an exact analytic solution expressed in terms of the bivariate Mittag-Leffler function, formulates several useful properties of this function to support the construction, and develops a numerical scheme based on sequential reformulation combined with the L1-finite element method.

Significance. If the exact solution holds rigorously for the stated problem class, the work would provide a useful closed-form representation for fractional models incorporating multiple memory effects, along with supporting properties of the bivariate Mittag-Leffler function and a practical discretization approach. The numerical component adds applicability, but the overall significance depends on whether the analytic claim is general or restricted by unstated conditions.

major comments (2)

Derivation of the exact analytic solution: The central claim is that the solution of the sequential Caputo problem ^C D^α (^C D^β u) = f(t) with the given boundary-initial conditions is exactly representable by the bivariate Mittag-Leffler function without remainder integrals or correction series. This representation is obtained via Laplace transform inversion, but the inversion produces precisely that form only when fractional-order compatibility relations hold between the initial data at t=0 and the boundary data at the other endpoint (e.g., the fractional integral of the boundary datum matching the initial datum up to order α). No such conditions are imposed or verified in the problem statement or solution construction, so the claimed direct representation does not hold for arbitrary data.
Numerical scheme section: The paper develops a numerical scheme via sequential reformulation and the L1-finite element method, yet provides no convergence analysis, stability estimates, or error bounds. Without these, it is impossible to assess whether the scheme reliably approximates the analytic solution or satisfies the boundary conditions to the expected order.

minor comments (1)

Abstract: The orders α and β of the sequential Caputo derivatives and the precise form of the boundary-initial conditions should be stated explicitly to allow readers to evaluate the scope of the exact-solution claim without consulting the full text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address each major comment below and outline the revisions we plan to make.

read point-by-point responses

Referee: Derivation of the exact analytic solution: The central claim is that the solution of the sequential Caputo problem ^C D^α (^C D^β u) = f(t) with the given boundary-initial conditions is exactly representable by the bivariate Mittag-Leffler function without remainder integrals or correction series. This representation is obtained via Laplace transform inversion, but the inversion produces precisely that form only when fractional-order compatibility relations hold between the initial data at t=0 and the boundary data at the other endpoint (e.g., the fractional integral of the boundary datum matching the initial datum up to order α). No such conditions are imposed or verified in the problem statement or solution construction, so the claimed direct representation does not hold for arbitrary data.

Authors: We appreciate the referee highlighting this subtlety in the solution representation. After reviewing the Laplace inversion procedure in our derivation, we recognize that compatibility conditions between the initial conditions and the boundary data are required to ensure the solution is precisely the bivariate Mittag-Leffler function without additional integral remainder terms. In the revised manuscript, we will incorporate these conditions into the problem statement, provide a verification that they are satisfied for the exact representation, and discuss the general case briefly. This will clarify the scope of the analytic result. revision: yes
Referee: Numerical scheme section: The paper develops a numerical scheme via sequential reformulation and the L1-finite element method, yet provides no convergence analysis, stability estimates, or error bounds. Without these, it is impossible to assess whether the scheme reliably approximates the analytic solution or satisfies the boundary conditions to the expected order.

Authors: We agree that the absence of a convergence analysis limits the assessment of the numerical scheme. We will revise the manuscript to include a detailed convergence analysis, including stability estimates and error bounds for the L1-finite element discretization. This will demonstrate the expected convergence rates and confirm that the scheme approximates the solution while respecting the boundary conditions. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard Laplace/series methods for sequential Caputo operators

full rationale

The paper states it derives the exact solution via techniques for sequential fractional operators and formulates bivariate Mittag-Leffler properties to support construction. No quoted step reduces a prediction to a fitted input, self-definition, or load-bearing self-citation chain. The representation follows directly from applying the Laplace transform to ^C D^α (^C D^β u) = f(t) and inverting under the given boundary-initial conditions, which is independent of the target result itself. External benchmarks (standard ML function identities) remain available and are not redefined within the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Caputo derivatives and Mittag-Leffler functions from prior literature; no new free parameters or invented entities are introduced beyond the problem setup.

axioms (2)

domain assumption Caputo fractional derivatives of order alpha satisfy the usual semigroup property under sequential application
Invoked to allow reformulation and solution via bivariate Mittag-Leffler
standard math The bivariate Mittag-Leffler function satisfies the required differential relations for the sequential operator
Used to construct the exact solution

pith-pipeline@v0.9.0 · 5382 in / 1204 out tokens · 24780 ms · 2026-05-09T21:34:38.389272+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

[1]

Hattaf, On the stability and numerical scheme of fractional differential equations with application to biology, Computation 10 (2022), 97, DOI: 10.3390/computation10060097

K. Hattaf, On the stability and numerical scheme of fractional differential equations with application to biology, Computation 10 (2022), 97, DOI: 10.3390/computation10060097

work page doi:10.3390/computation10060097 2022
[2]

Traore and N

A. Traore and N. Sene, Model of economic growth in the context of fractional derivative, Alexandria Engineering Journal 59 (2020), 4843--4850, DOI: 10.1016/j.aej.2020.08.047

work page doi:10.1016/j.aej.2020.08.047 2020
[3]

Cesarone, M

F. Cesarone, M. Caputo and C. Cametti, Memory formalism in the passive diffusion across highly heterogeneous systems, Journal of Membrane Science 250 (2005), 79--84, DOI: 10.1016/j.memsci.2004.10.018

work page doi:10.1016/j.memsci.2004.10.018 2005
[4]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley Sons, New York, 1993

work page 1993
[5]

Ahmad and J

B. Ahmad and J. J. Nieto, Sequential fractional differential equations with three-point boundary conditions, Computers Mathematics with Applications 64 (2012), 3046--3052, DOI:10.1016/j.camwa.2012.02.036

work page doi:10.1016/j.camwa.2012.02.036 2012
[6]

Ahmad and J

B. Ahmad and J. J. Nieto, Boundary value problems for a class of sequential integrodifferential equations of fractional order, Journal of Function Spaces and Applications 2013 (2013), Art. ID 149659, DOI:10.1155/2013/149659

work page doi:10.1155/2013/149659 2013
[7]

Ahmad and S

B. Ahmad and S. K. Ntouyas, On higher-order sequential fractional differential inclusions with nonlocal three-point boundary conditions, Abstract and Applied Analysis 2014 (2014), Art. ID 659405, DOI: 10.1155/2014/659405

work page doi:10.1155/2014/659405 2014
[8]

Mozyrska, E

D. Mozyrska, E. Girejko and M. Wyrwas, Fractional nonlinear systems with sequential operators, Central European Journal of Physics 11 (2013), 1295--1303, DOI: 10.2478/s11534-013-0223-3

work page doi:10.2478/s11534-013-0223-3 2013
[9]

Song and Y

S. Song and Y. Cui, Existence of solutions for integral boundary value problems of mixed fractional differential equations under resonance, Boundary Value Problems 2020 (2020), 23, DOI: 10.1186/s13661-020-01332-5

work page doi:10.1186/s13661-020-01332-5 2020
[10]

K. M. Furati, Bounds on the solution of a Cauchy-type problem involving a weighted sequential fractional derivative, Fractional Calculus and Applied Analysis 16 (2013), 171--188, DOI:10.2478/s13540-013-0012-0

work page doi:10.2478/s13540-013-0012-0 2013
[11]

A. M. Djaouti, K. O. Melha and M. A. Latif, New results on the solvability of abstract sequential Caputo fractional differential equations with a resolvent-operator approach and applications, Mathematics 12 (2024), 1268, DOI: 10.3390/math12081268

work page doi:10.3390/math12081268 2024
[12]

Yu. E. Fayziev and Sh. F. Jumaeva, On the Cauchy problem for the Langevin-type fractional equation, Uzbek Mathematical Journal 69 (2025), 109--118

work page 2025
[13]

Yu. E. Fayziev, Sh. F. Jumaeva and F. Abdullaeva, Inverse problem for the Langevin-type fractional differential equation, Bulletin of the Institute of Mathematics 8 (2025), 30--35

work page 2025
[14]

Fayziev and Sh

Yu. Fayziev and Sh. Jumaeva, Forward and Inverse Problems for a Langevin-Type Fractional Equation Involving Non-Local Time Condition, preprint (2025), https://arxiv.org/pdf/2507.07446

work page arXiv 2025
[15]

Boichuk and V

O. Boichuk and V. Feruk, Fredholm Boundary-Value Problem for the Two-Term Sequential Fractional Differential Equation, Ukrainian Mathematical Journal 77 (2025), 1--11, DOI: 10.3842/umzh.v77i1.8611

work page doi:10.3842/umzh.v77i1.8611 2025
[16]

Ashurov and R

R. Ashurov and R. Saparbayev, Fractional Telegraph Equation with the Caputo Derivative, Fractal and Fractional 7 (2023), 483, DOI: 10.3390/fractalfract7060483

work page doi:10.3390/fractalfract7060483 2023
[17]

Ashurov and R

R. Ashurov and R. Saparbayev, Forward and Inverse Problems for Fractional Telegraph Equation, Lobachevskii Journal of Mathematics 45 (2024), 4459--4478, DOI: 10.1134/S199508022460506X

work page doi:10.1134/s199508022460506x 2024
[18]

Saparbayev, A Time-non-local problem for the fractional-order telegraph equation, Bulletin of the Institute of Mathematics 7 (2024), 63--76

R. Saparbayev, A Time-non-local problem for the fractional-order telegraph equation, Bulletin of the Institute of Mathematics 7 (2024), 63--76

work page 2024
[19]

Ashurov and R

R. Ashurov and R. Saparbayev, Time-dependent identification problem for a fractional Telegraph equation with the Caputo derivative, Fractional Calculus and Applied Analysis 27 (2024), 652--676, DOI: 10.1007/s13540-024-00240-0

work page doi:10.1007/s13540-024-00240-0 2024
[20]

R. R. Ashurov and R. A. Saparbayev, Inverse Problem for Determining Time-Dependent Coefficient and Source Functions in a Time-Fractional Telegraph Equation, Lobachevskii Journal of Mathematics, to appear

work page
[21]

Lizama, Abstract linear fractional evolution equations, in: Handbook of Fractional Calculus with Applications 2, De Gruyter (2019), 465--498, DOI: 10.1515/9783110571660-021

C. Lizama, Abstract linear fractional evolution equations, in: Handbook of Fractional Calculus with Applications 2, De Gruyter (2019), 465--498, DOI: 10.1515/9783110571660-021

work page doi:10.1515/9783110571660-021 2019
[22]

Ashurov and M

R. Ashurov and M. Shakarova, Inverse problem for subdiffusion equation with the integral over-determination condition, Journal of Mathematical Sciences (2024), DOI: 10.13108/2024-16-1-112

work page doi:10.13108/2024-16-1-112 2024
[23]

M. M. Djrbashian, Integral Transforms and Representation of Functions in the Complex Domain, Nauka, Moscow, 1966

work page 1966
[24]

T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Mathematical Journal 19 (1971), 7--15

work page 1971
[25]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, The Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006

work page 2006
[26]

M. Garg, P. Manohar and S. L. Kalla, A Mittag-Leffler-type function of two variables, Integral Transforms and Special Functions 24 (2013), 934--944, DOI:10.1080/10652469.2013.789872

work page doi:10.1080/10652469.2013.789872 2013
[27]

E. T. Karimov and A. Hasanov, On a boundary-value problem in a bounded domain for a time-fractional diffusion equation with the Prabhakar fractional derivative, Bulletin of the Karaganda University Mathematics 111 (2023), 39--46, DOI:10.31489/2023m3/39-46

work page doi:10.31489/2023m3/39-46 2023
[28]

Karimov and M

E. Karimov and M. Toshpulatov, Mixed wave-diffusion-wave equation: solvability of an initial-boundary problem, Gulf Journal of Mathematics 20 (2025), 120--139, DOI:10.56947/gjom.v20i.2863

work page doi:10.56947/gjom.v20i.2863 2025
[29]

Karimov, N

E. Karimov, N. Tokmagambetov and M. Toshpulatov, On a Mixed Equation Involving Prabhakar Fractional Order Integral-Differential Operators, in: Extended Abstracts 2021/2022, APDEGS, Springer

work page 2021
[30]

Hasanov and H

A. Hasanov and H. Yuldashova, Mittag-Leffler type functions of three variables, Mathematical Methods in the Applied Sciences (2024), DOI:10.1002/mma.10401

work page doi:10.1002/mma.10401 2024
[31]

Fernandez, C

A. Fernandez, C. Kürt and M. A. Özarslan, A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators, Computational and Applied Mathematics 39 (2020), 200, DOI:10.1007/s40314-020-01224-5

work page doi:10.1007/s40314-020-01224-5 2020
[32]

Maes and K

F. Maes and K. Van Bockstal, Existence and uniqueness of a weak solution to fractional single-phase-lag heat equation, Fractional Calculus and Applied Analysis 26 (2023), 1663--1690, DOI:10.1007/s13540-023-00177-w

work page doi:10.1007/s13540-023-00177-w 2023
[33]

Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010

work page 2010

[1] [1]

Hattaf, On the stability and numerical scheme of fractional differential equations with application to biology, Computation 10 (2022), 97, DOI: 10.3390/computation10060097

K. Hattaf, On the stability and numerical scheme of fractional differential equations with application to biology, Computation 10 (2022), 97, DOI: 10.3390/computation10060097

work page doi:10.3390/computation10060097 2022

[2] [2]

Traore and N

A. Traore and N. Sene, Model of economic growth in the context of fractional derivative, Alexandria Engineering Journal 59 (2020), 4843--4850, DOI: 10.1016/j.aej.2020.08.047

work page doi:10.1016/j.aej.2020.08.047 2020

[3] [3]

Cesarone, M

F. Cesarone, M. Caputo and C. Cametti, Memory formalism in the passive diffusion across highly heterogeneous systems, Journal of Membrane Science 250 (2005), 79--84, DOI: 10.1016/j.memsci.2004.10.018

work page doi:10.1016/j.memsci.2004.10.018 2005

[4] [4]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley Sons, New York, 1993

work page 1993

[5] [5]

Ahmad and J

B. Ahmad and J. J. Nieto, Sequential fractional differential equations with three-point boundary conditions, Computers Mathematics with Applications 64 (2012), 3046--3052, DOI:10.1016/j.camwa.2012.02.036

work page doi:10.1016/j.camwa.2012.02.036 2012

[6] [6]

Ahmad and J

B. Ahmad and J. J. Nieto, Boundary value problems for a class of sequential integrodifferential equations of fractional order, Journal of Function Spaces and Applications 2013 (2013), Art. ID 149659, DOI:10.1155/2013/149659

work page doi:10.1155/2013/149659 2013

[7] [7]

Ahmad and S

B. Ahmad and S. K. Ntouyas, On higher-order sequential fractional differential inclusions with nonlocal three-point boundary conditions, Abstract and Applied Analysis 2014 (2014), Art. ID 659405, DOI: 10.1155/2014/659405

work page doi:10.1155/2014/659405 2014

[8] [8]

Mozyrska, E

D. Mozyrska, E. Girejko and M. Wyrwas, Fractional nonlinear systems with sequential operators, Central European Journal of Physics 11 (2013), 1295--1303, DOI: 10.2478/s11534-013-0223-3

work page doi:10.2478/s11534-013-0223-3 2013

[9] [9]

Song and Y

S. Song and Y. Cui, Existence of solutions for integral boundary value problems of mixed fractional differential equations under resonance, Boundary Value Problems 2020 (2020), 23, DOI: 10.1186/s13661-020-01332-5

work page doi:10.1186/s13661-020-01332-5 2020

[10] [10]

K. M. Furati, Bounds on the solution of a Cauchy-type problem involving a weighted sequential fractional derivative, Fractional Calculus and Applied Analysis 16 (2013), 171--188, DOI:10.2478/s13540-013-0012-0

work page doi:10.2478/s13540-013-0012-0 2013

[11] [11]

A. M. Djaouti, K. O. Melha and M. A. Latif, New results on the solvability of abstract sequential Caputo fractional differential equations with a resolvent-operator approach and applications, Mathematics 12 (2024), 1268, DOI: 10.3390/math12081268

work page doi:10.3390/math12081268 2024

[12] [12]

Yu. E. Fayziev and Sh. F. Jumaeva, On the Cauchy problem for the Langevin-type fractional equation, Uzbek Mathematical Journal 69 (2025), 109--118

work page 2025

[13] [13]

Yu. E. Fayziev, Sh. F. Jumaeva and F. Abdullaeva, Inverse problem for the Langevin-type fractional differential equation, Bulletin of the Institute of Mathematics 8 (2025), 30--35

work page 2025

[14] [14]

Fayziev and Sh

Yu. Fayziev and Sh. Jumaeva, Forward and Inverse Problems for a Langevin-Type Fractional Equation Involving Non-Local Time Condition, preprint (2025), https://arxiv.org/pdf/2507.07446

work page arXiv 2025

[15] [15]

Boichuk and V

O. Boichuk and V. Feruk, Fredholm Boundary-Value Problem for the Two-Term Sequential Fractional Differential Equation, Ukrainian Mathematical Journal 77 (2025), 1--11, DOI: 10.3842/umzh.v77i1.8611

work page doi:10.3842/umzh.v77i1.8611 2025

[16] [16]

Ashurov and R

R. Ashurov and R. Saparbayev, Fractional Telegraph Equation with the Caputo Derivative, Fractal and Fractional 7 (2023), 483, DOI: 10.3390/fractalfract7060483

work page doi:10.3390/fractalfract7060483 2023

[17] [17]

Ashurov and R

R. Ashurov and R. Saparbayev, Forward and Inverse Problems for Fractional Telegraph Equation, Lobachevskii Journal of Mathematics 45 (2024), 4459--4478, DOI: 10.1134/S199508022460506X

work page doi:10.1134/s199508022460506x 2024

[18] [18]

Saparbayev, A Time-non-local problem for the fractional-order telegraph equation, Bulletin of the Institute of Mathematics 7 (2024), 63--76

R. Saparbayev, A Time-non-local problem for the fractional-order telegraph equation, Bulletin of the Institute of Mathematics 7 (2024), 63--76

work page 2024

[19] [19]

Ashurov and R

R. Ashurov and R. Saparbayev, Time-dependent identification problem for a fractional Telegraph equation with the Caputo derivative, Fractional Calculus and Applied Analysis 27 (2024), 652--676, DOI: 10.1007/s13540-024-00240-0

work page doi:10.1007/s13540-024-00240-0 2024

[20] [20]

R. R. Ashurov and R. A. Saparbayev, Inverse Problem for Determining Time-Dependent Coefficient and Source Functions in a Time-Fractional Telegraph Equation, Lobachevskii Journal of Mathematics, to appear

work page

[21] [21]

Lizama, Abstract linear fractional evolution equations, in: Handbook of Fractional Calculus with Applications 2, De Gruyter (2019), 465--498, DOI: 10.1515/9783110571660-021

C. Lizama, Abstract linear fractional evolution equations, in: Handbook of Fractional Calculus with Applications 2, De Gruyter (2019), 465--498, DOI: 10.1515/9783110571660-021

work page doi:10.1515/9783110571660-021 2019

[22] [22]

Ashurov and M

R. Ashurov and M. Shakarova, Inverse problem for subdiffusion equation with the integral over-determination condition, Journal of Mathematical Sciences (2024), DOI: 10.13108/2024-16-1-112

work page doi:10.13108/2024-16-1-112 2024

[23] [23]

M. M. Djrbashian, Integral Transforms and Representation of Functions in the Complex Domain, Nauka, Moscow, 1966

work page 1966

[24] [24]

T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Mathematical Journal 19 (1971), 7--15

work page 1971

[25] [25]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, The Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006

work page 2006

[26] [26]

M. Garg, P. Manohar and S. L. Kalla, A Mittag-Leffler-type function of two variables, Integral Transforms and Special Functions 24 (2013), 934--944, DOI:10.1080/10652469.2013.789872

work page doi:10.1080/10652469.2013.789872 2013

[27] [27]

E. T. Karimov and A. Hasanov, On a boundary-value problem in a bounded domain for a time-fractional diffusion equation with the Prabhakar fractional derivative, Bulletin of the Karaganda University Mathematics 111 (2023), 39--46, DOI:10.31489/2023m3/39-46

work page doi:10.31489/2023m3/39-46 2023

[28] [28]

Karimov and M

E. Karimov and M. Toshpulatov, Mixed wave-diffusion-wave equation: solvability of an initial-boundary problem, Gulf Journal of Mathematics 20 (2025), 120--139, DOI:10.56947/gjom.v20i.2863

work page doi:10.56947/gjom.v20i.2863 2025

[29] [29]

Karimov, N

E. Karimov, N. Tokmagambetov and M. Toshpulatov, On a Mixed Equation Involving Prabhakar Fractional Order Integral-Differential Operators, in: Extended Abstracts 2021/2022, APDEGS, Springer

work page 2021

[30] [30]

Hasanov and H

A. Hasanov and H. Yuldashova, Mittag-Leffler type functions of three variables, Mathematical Methods in the Applied Sciences (2024), DOI:10.1002/mma.10401

work page doi:10.1002/mma.10401 2024

[31] [31]

Fernandez, C

A. Fernandez, C. Kürt and M. A. Özarslan, A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators, Computational and Applied Mathematics 39 (2020), 200, DOI:10.1007/s40314-020-01224-5

work page doi:10.1007/s40314-020-01224-5 2020

[32] [32]

Maes and K

F. Maes and K. Van Bockstal, Existence and uniqueness of a weak solution to fractional single-phase-lag heat equation, Fractional Calculus and Applied Analysis 26 (2023), 1663--1690, DOI:10.1007/s13540-023-00177-w

work page doi:10.1007/s13540-023-00177-w 2023

[33] [33]

Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010

work page 2010