## Publications

317 results found

Ruzhansky M, Torebek BT, 2021, Van der Corput lemmas for Mittag-Leffler functions. II. α–directions, *Bulletin des Sciences Mathematiques*, Vol: 171, ISSN: 0007-4497

The paper is devoted to study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory integrals appearing in the analysis of time-fractional partial differential equations. More specifically, we study integral of the form Iα,β(λ)=∫REα,β(iαλϕ(x))ψ(x)dx, for the range 0<α≤2,β>0. This extends the variety of estimates obtained in the first part, where integrals with functions Eα,β(iλϕ(x)) have been studied. Several generalisations of the van der Corput lemmas are proved. As an application of the above results, the generalised Riemann-Lebesgue lemma, the Cauchy problem for the time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

Chatzakou M, Delgado J, Ruzhansky M, 2021, On aclass of anharmonic oscillators, *JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES*, Vol: 153, Pages: 1-29, ISSN: 0021-7824

Restrepo JE, Ruzhansky M, Suragan D, 2021, Explicit solutions for linear variable–coefficient fractional differential equations with respect to functions, *Applied Mathematics and Computation*, Vol: 403, ISSN: 0096-3003

Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite series of fractional integro-differential operators, which can be widely and efficiently used for analytic and computational purposes. In the case of constant coefficients, the solution can be expressed in terms of the multivariate Mittag-Leffler functions. In particular, the obtained result extends the Luchko-Gorenflo representation formula [1, Theorem 4.1] to a general class of linear fractional differential equations with variable coefficients, to complex fractional derivatives, and to fractional derivatives with respect to a given function.

Ruzhansky M, Velasquez-Rodriguez JP, 2021, Non-harmonic Gohberg's lemma, Gershgorin theory and heat equation on manifolds with boundary, *MATHEMATISCHE NACHRICHTEN*, ISSN: 0025-584X

Dasgupta A, Ruzhansky M, 2021, Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality (vol 27, 15, 2021), *JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS*, Vol: 27, ISSN: 1069-5869

Ruzhansky M, Verma D, 2021, Hardy inequalities on metric measure spaces, II: the case p > q, *PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, Vol: 477, ISSN: 1364-5021

Altybay A, Ruzhansky M, Sebih ME,
et al., 2021, The heat equation with strongly singular potentials, *Applied Mathematics and Computation*, Vol: 399, Pages: 1-15, ISSN: 0096-3003

In this paper we consider the heat equation with strongly singular potentials and prove that it has a ”very weak solution”. Moreover, we show the uniqueness and consistency results in some appropriate sense. The cases of positive and negative potentials are studied. Numerical simulations are done: one suggests so-called ”laser heating and cooling” effects depending on a sign of the potential. The latter is justified by the physical observations.

Kumar V, Ruzhansky M, 2021, A note on K-functional, Modulus of smoothness, Jackson theorem and Bernstein–Nikolskii–Stechkin inequality on Damek–Ricci spaces, *Journal of Approximation Theory*, Vol: 264, ISSN: 0021-9045

In this paper we study approximation theorems for L2-space on Damek–Ricci spaces. We prove direct Jackson theorem of approximations for the modulus of smoothness defined using spherical mean operator on Damek–Ricci spaces. We also prove Bernstein–Nikolskii–Stechkin inequality. To prove these inequalities we use functions of bounded spectrum as a tool of approximation. Finally, as an application we prove equivalence of the K-functional and modulus of smoothness for Damek–Ricci spaces.

Dasgupta A, Ruzhansky M, 2021, Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality, *Journal of Fourier Analysis and Applications*, Vol: 27, ISSN: 1069-5869

In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our papers (Dasgupta and Ruzhansky in Trans Am Math Soc 368(12):8481–8498, 2016) and (Dasgupta and Ruzhansky in Trans Am Math Soc Ser B 5:81–101, 2018). We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary.

Ruzhansky M, Taranto CA, 2021, Time-Dependent Wave Equations on Graded Groups, *ACTA APPLICANDAE MATHEMATICAE*, Vol: 171, ISSN: 0167-8019

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- Citations: 1

Altybay A, Ruzhansky M, Sebih ME,
et al., 2021, Fractional Schrödinger Equation with Singular Potentials of Higher Order, *Reports on Mathematical Physics*, Vol: 87, Pages: 129-144, ISSN: 0034-4877

In this paper the space-fractional Schrödinger equations with singular potentials are studied. Delta like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of ‘weakened’ solutions, calling them very weak solutions. The existence, uniqueness and consistency results are proved in an appropriate sense. Numerical simulations are done, and a particles accumulating effect is observed in the singular cases. From the mathematical point of view a “splitting of the strong singularity” phenomena is also observed.

Altybay A, Ruzhansky M, Sebih ME,
et al., 2021, Fractional Klein-Gordon equation with singular mass, *Chaos, Solitons and Fractals*, Vol: 143, ISSN: 0960-0779

We consider a space-fractional wave equation with a singular mass term depending on the position and prove that it is very weak well-posed. The uniqueness is proved in some appropriate sense. Moreover, we prove the consistency of the very weak solution with classical solutions when they exist. In order to study the behaviour of the very weak solution near the singularities of the coefficient, some numerical experiments are conducted where the appearance of a wall effect for the singular masses of the strength of δ2 is observed.

Kirilov A, de Moraes WAA, Ruzhansky M, 2021, Global hypoellipticity and global solvability for vector fields on compact Lie groups, *JOURNAL OF FUNCTIONAL ANALYSIS*, Vol: 280, ISSN: 0022-1236

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- Citations: 2

Bin-Saad MG, Hasanov A, Ruzhansky M, 2021, Some properties relating to the Mittag–Leffler function of two variables, *Integral Transforms and Special Functions*, ISSN: 1065-2469

An attempt is made here to study the Mittag–Leffler function with two variables. Its various properties including integral and operational relationships with other known Mittag–Leffler functions of one variable, pure and differential recurrence relations, Euler transform, Laplace transform, Mellin transform, Whittaker transform, Mellin–Barnes integral representation, and its relationship with Wright hypergeometric function are investigated and established. Also, properties of the Mittag–Leffler function of two variables associated with fractional calculus operators are considered.

Cardona D, Delgado J, Ruzhansky M, 2021, L<sup>p</sup> -Bounds for Pseudo-differential Operators on Graded Lie Groups, *Journal of Geometric Analysis*, ISSN: 1050-6926

In this work we obtain sharp Lp-estimates for pseudo-differential operators on arbitrary graded Lie groups. The results are presented within the setting of the global symbolic calculus on graded Lie groups by using the Fourier analysis associated to every graded Lie group which extends the usual one due to Hörmander on Rn. The main result extends the classical Fefferman’s sharp theorem on the Lp-boundedness of pseudo-differential operators for Hörmander classes on Rn to general graded Lie groups, also adding the borderline ρ= δ case.

Fischer V, Ruzhansky M, 2021, Fourier multipliers on graded lie groups, *Colloquium Mathematicum*, Vol: 165, Pages: 1-30, ISSN: 0010-1354

We study multipliers on graded nilpotent Lie groups defined via group Fourier transform. More precisely, we show that Hörmander-type conditions on the Fourier multipliers imply Lp-boundedness. We express these conditions using difference operators and positive Rockland operators. We also obtain a more refined condition using Sobolev spaces on the dual of the group which are defined and studied in this paper.

Karimov E, Ruzhansky M, Tokmagambetov N, 2021, Cauchy type problems for fractional differential equations, *Integral Transforms and Special Functions*, ISSN: 1065-2469

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann–Liouville derivatives is less understood. In this paper, we propose new type initial, inner, and inner-boundary value problems for fractional differential equations with the Riemann–Liouville derivatives. The results on the existence and uniqueness are proved, and conditions on the solvability are found. The well-posedness of the new type of initial, inner, and inner-boundary conditions is also discussed. Moreover, we give explicit formulas for the solutions. As an application fractional partial differential equations for general positive operators are studied.

Ruzhansky M, Sabitbek B, Suragan D, 2021, Principal frequency of p-sub-Laplacians for general vector fields, *Zeitschrift fur Analysis und ihre Anwendung*, Vol: 40, Pages: 97-109, ISSN: 0232-2064

In this paper, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet p-sub-Laplacian for general vector fields. As a byproduct, we establish the Caccioppoli inequalities and also discuss the particular cases on the Grushin plane and on the Heisenberg group.

Kassymov A, Ruzhansky M, Tokmagambetov N,
et al., 2021, Sobolev, Hardy, Gagliardo–Nirenberg, and Caffarelli–Kohn–Nirenberg-type inequalities for some fractional derivatives, *Banach Journal of Mathematical Analysis*, Vol: 15, ISSN: 1735-8787

In this paper, we show different inequalities for fractional-order differential operators. In particular, the Sobolev, Hardy, Gagliardo–Nirenberg, and Caffarelli–Kohn–Nirenberg-type inequalities for the Caputo, Riemann–Liouville, and Hadamard derivatives are obtained. In addition, we show some applications of these inequalities.

Ruzhansky M, Serikbaev D, Torebek BT,
et al., 2021, Direct and inverse problems for time-fractional pseudo-parabolic equations, *Quaestiones Mathematicae*, ISSN: 1607-3606

The purpose of this paper is to establish the solvability results to direct and inverse problems for time-fractional pseudo-parabolic equations with the self-adjoint operators. We are especially interested in proving existence and uniqueness of the solutions in the abstract setting of Hilbert spaces.

Ruzhansky M, Sabitbek B, Suragan D, 2021, Geometric Hardy Inequalities on Starshaped Sets, *Journal of Convex Analysis*, Vol: 28, ISSN: 0944-6532

We present geometric Hardy inequalities on starshaped sets in Carnot groups. Also, we obtain geometric Hardy inequalities on half-spaces for general vector fields.

Avetisyan Z, Grigoryan M, Ruzhansky M, 2021, Approximations in L<sup>1</sup> with convergent Fourier series, *Mathematische Zeitschrift*, ISSN: 0025-5874

For a separable finite diffuse measure space M and an orthonormal basis { φn} of L2(M) consisting of bounded functions φn∈ L∞(M) , we find a measurable subset E⊂ M of arbitrarily small complement | M\ E| < ϵ, such that every measurable function f∈ L1(M) has an approximant g∈ L1(M) with g= f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of M= G/ H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

Delgado J, Ruzhansky M, 2021, Schatten-von Neumann classes of integral operators, *Journal des Mathematiques Pures et Appliquees*, ISSN: 0021-7824

In this work we establish sharp kernel conditions ensuring that the corresponding integral operators belong to Schatten-von Neumann classes. The conditions are given in terms of the spectral properties of operators acting on the kernel. As applications we establish several criteria in terms of different types of differential operators and their spectral asymptotics in different settings: compact manifolds, operators on lattices, domains in Rn of finite measure, and conditions for operators on Rn given in terms of anharmonic oscillators. We also give examples in the settings of compact sub-Riemannian manifolds, contact manifolds, strictly pseudo-convex CR manifolds, and (sub-)Laplacians on compact Lie groups.

Ashyralyev A, Kalmenov TS, Ruzhansky MV, et al., 2021, Preface, Pages: v-vi, ISSN: 2194-1009

Ruzhansky M, Yessirkegenov N, 2021, Critical Gagliardo-Nirenberg, Trudinger, Brezis-Gallouet-Wainger inequalities on graded groups and ground states, *Communications in Contemporary Mathematics*, ISSN: 0219-1997

In this paper, we investigate critical Gagliardo-Nirenberg, Trudinger-type and Brezis-Gallouet-Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which include the cases of ℝn, Heisenberg, and general stratified Lie groups. As an application, using the critical Gagliardo-Nirenberg inequality, the existence of least energy solutions of nonlinear Schrödinger type equations is obtained. We also express the best constant in the critical Gagliardo-Nirenberg and Trudinger inequalities in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The obtained results are already new in the setting of general stratified Lie groups (homogeneous Carnot groups). Among new technical methods, we also extend Folland's analysis of Hölder spaces from stratified Lie groups to general homogeneous Lie groups.

Fischer V, Ruzhansky M, Taranto CA, 2020, Subelliptic Gevrey spaces, Publisher: WILEY-V C H VERLAG GMBH

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- Citations: 1

Altybay A, Ruzhansky M, Tokmagambetov N, 2020, A parallel hybrid implementation of the 2D acoustic wave equation, *International Journal of Nonlinear Sciences and Numerical Simulation*, Vol: 21, Pages: 821-827, ISSN: 1565-1339

In this paper, we propose a hybrid parallel programming approach for a numerical solution of a two-dimensional acoustic wave equation using an implicit difference scheme for a single computer. The calculations are carried out in an implicit finite difference scheme. First, we transform the differential equation into an implicit finite-difference equation and then using the alternating direction implicit (ADI) method, we split the equation into two sub-equations. Using the cyclic reduction algorithm, we calculate an approximate solution. Finally, we change this algorithm to parallelize on graphics processing unit (GPU), GPU + Open Multi-Processing (OpenMP), and Hybrid (GPU + OpenMP + message passing interface (MPI)) computing platforms. The special focus is on improving the performance of the parallel algorithms to calculate the acceleration based on the execution time. We show that the code that runs on the hybrid approach gives the expected results by comparing our results to those obtained by running the same simulation on a classical processor core, Compute Unified Device Architecture (CUDA), and CUDA + OpenMP implementations.

Ruzhansky M, Torebek BT, 2020, Multidimensional van der Corput-Type estimates involving Mittag-Leffler functions, *Fractional Calculus and Applied Analysis*, Vol: 23, Pages: 1663-1677, ISSN: 1311-0454

The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with functions Eα, β(i λ φ(x)), x ϵ ℝN and Eα, β(iα λ φ(x)), x ϵ ℝN for the various range of α and β. Several generalisations of the van der Corput-type estimates are proved. As an application of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

Ruzhansky M, Sabitbek B, Suragan D, 2020, Geometric Hardy and Hardy-Sobolev inequalities on Heisenberg groups, *BULLETIN OF MATHEMATICAL SCIENCES*, Vol: 10, ISSN: 1664-3607

Garetto C, Jäh C, Ruzhansky M, 2020, Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, II: Microlocal analysis, *Journal of Differential Equations*, Vol: 269, Pages: 7881-7905, ISSN: 0022-0396

In this paper we continue the study of non-diagonalisable hyperbolic systems with variable multiplicity started by the authors in [1]. In the case of space dependent coefficients, we prove a representation formula for solutions that allows us to derive results of regularity and propagation of singularities.

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