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Niyaz Tokmagambetov

Publications and source records attributed to Niyaz Tokmagambetov.

At least 19 recordsLinked to original sources
arXiv2026

Moduli of smoothness and growth of the Laguerre transform: two-sided estimates

We study weighted coefficient estimates for the Laguerre transform in terms of moduli of smoothness generated by the Laguerre translation. For every integer $r\ge1$ and $1\le p\le2$, a weighted $\ell^{p'}$ norm of $\min\{1,nt\}^{r}\widehat f_α(n)$ is bounded by the modulus of order $r$; for $2\le p\le\infty$ the reverse direction holds on the same weighted sequence scale. At $p=2$ these estimates yield a two-sided equivalence, and testing individual Laguerre polynomials proves optimality of the multiplier weight up to constants. The proof combines Hausdorff--Young inequalities with an averaged lower bound for the translation multiplier. As consequences we recover Jackson- and Bernstein-type estimates and the known $L^2$ approximation characterisation of the Lipschitz classes for $0<β<r$. We also give the corresponding one-sided coefficient-tail criteria for general $p$, together with saturation and endpoint statements.

math.CA↗
arXiv2026

Power-kernel fractional Sturm--Liouville operators: a graph realization and boundary-independent singular-value asymptotics

For $\tfrac12<α<1$ we construct a closed realization of $\mathcal L u=\mathcal D_1^αD_0^αu$ in $L^2(0,1)$ whose domain contains the singular mode $x^{α-1}$. The domain, defined via the distributional Caputo composition, is characterized by $I_0^{1-α}u\in H^1(0,1)$ and $D_0^αu-c\in\operatorname{ran}I_1^α$; we prove a Volterra representation for its elements, graph-norm completeness, and that $\mathcal L$ is the adjoint of its zero-trace restriction. Four bounded endpoint traces give an ordinary boundary triplet, so all self-adjoint extensions correspond to Lagrangian planes in $\mathbb C^4$, including coupled endpoint conditions. For each we give an exact zero-eigenvalue criterion and, when $0\inρ(\mathcal L_ω)$, an inverse of the form $I_0^αI_1^α$ plus an explicit finite-rank correction. Sharp singular-value asymptotics $s_n(\mathcal L_ω^{-1})=(πn)^{-2α}\bigl(1+O(n^{-1})\bigr)$ yield $\mathcal L_ω^{-1}\in\mathfrak S_p\iff p>1/(2α)$, with weak-Schatten endpoint and Weyl asymptotics --- now for \emph{every} invertible boundary plane, not only the positive reference problem. As an application we obtain a mild-solution theorem for a Caputo-time diffusion equation in $C([0,T];L^2(0,1))$.

math.SP↗
arXiv2026

Dilation-balanced product Pitt inequalities for mixed-tail potentials

Motivated by the one-dimensional and radial theory of general monotone functions, we introduce a natural multiparameter general-monotonicity class for product Fourier inequalities. The monotonicity condition is replaced by an intrinsic mixed-tail condition: a function on the positive orthant is the upper-tail potential of a complex Radon measure, and the integrated total variation of this measure is controlled by a local integral of the function. The representing measure is recovered as the full mixed distributional derivative, so the condition is intrinsic rather than coordinatewise. For coordinatewise even extensions we prove the anisotropic mixed-norm estimate \[ \|Π_α\widehat f_{A}\|_{L_{\vec q}(\mathbb R^d)} \le C\|Π_βf\|_{L_{\vec p}(\mathbb R^d)}, \qquad β_k=1-\frac1{p_k}-\frac1{q_k}-α_k, \] where $1\le p_k\le q_k<\infty$ and $α_k>-1/q_k$. Here $\widehat f_A$ is an Abel-summed Fourier transform and agrees almost everywhere with the ordinary Fourier transform for $f\in L^1$. The exponent relation is forced by independent coordinate dilations. Thus the theorem has the same coordinatewise dilation balance as the anisotropic monotone theory, while the hypotheses are formulated in the spirit of general monotonicity: variation is controlled locally, and no product or coordinatewise monotonicity structure is imposed. The class contains all sectorial mixed-tail potentials, an infinite-dimensional cone generated by arbitrary non-product positive measures satisfying the mixed-moment condition, and real nonseparable functions which are not monotone in the coordinate variables. For compactly supported examples which are bounded below by a positive constant in a neighborhood of the origin, the range $-1/q_k<α_k<1-1/q_k$ is exactly the range in which both weighted norms are finite.

math.CA↗
arXiv2026

Late-Time Fractional-Order Identification in Caputo Diffusion Equation

We study late-time identification of the Caputo order in a linear diffusion equation generated by a strictly positive self-adjoint operator with compact resolvent. For signed scalar observations \(M_α(t)=\sum_n a_nE_{α,1}(-λ_nt^α)\) satisfying \(\sum_n|a_n|/λ_n<\infty\), we show that, after eigenspace grouping, every nontrivial observation has a finite first nonzero resolvent moment \(S_m=\sum_n a_n/λ_n^m\). A uniform differentiated large-argument expansion of the Mittag-Leffler factor yields eventual strict monotonicity of \(α\mapsto M_α(t)\) on admissible intervals avoiding the zeros of \(1/Γ(1-mα)\), hence uniqueness from one sufficiently late scalar measurement. For two measurements, \(M_α(ρt)/M_α(t)=ρ^{-mα}(1+O(t^{-α_0}))\), giving a log-ratio estimator with asymptotic-bias and relative-noise error bounds. For bounded observations, \(S_m=\langle\mathcal A^{-m}φ,h\rangle\); for a finite rod, the leading point-sensor condition is \((\mathcal A^{-1}φ)(x_*)\ne0\). Counterexamples show the sharpness of the exclusions and noise interpretation.

math.AP↗
arXiv2026

On a fractional nonlinear Schrödinger equation with irregular coefficients. case: d<2s

In the case when $d<2s$, where $d$ is the space dimension and $s$ is the fractional power of the Laplacian, we study the well-posedness for a cubic nonlinear Schrödinger equation (CNLSE) generated by the fractional Laplacian and involving distributional, or less regular, coefficients. We formulate our problem in the setting of the concept of so-called very weak solutions and prove that it has a very weak solution. Moreover, we prove the uniqueness in some adequate sense as well as the compatibility of the very weak solution with the classical one when the latter exists. Our results cover the classical case when: $d=1, s=1$. A second task in this paper is to conduct some numerical experiments where interesting behaviours of the very weak solution are observed. The obtained result is the first example of the very weak well-posedness in the setting of nonlinear partial differential equations.

math.AP↗
arXiv2023

Fractional wave equation with irregular mass and dissipation

In this paper, we pursue our series of papers aiming to show the applicability of the concept of very weak solutions. We consider a wave model with irregular position dependent mass and dissipation terms, in particular, allowing for delta-like coefficients and prove that the problem has a very weak solution. Furthermore, we prove the uniqueness in an appropriate sense and the coherence of the very weak solution concept with classical theory. A special case of the model considered here, is the so-called telegraph equation.

math.AP↗
arXiv2023

One inverse source problem generated by the Dunkl operator

The aim of this paper is to study time-fractional pseudo-parabolic type equations generated by the Dunkl operator. The forward problem is considered and its well-posedness is established. In particular, a prior estimates are obtained in the Sobolev type spaces and, explicit formulas for solutions of the problems are derived. Here we also deal with the left-sided Caputo fractional time derivative. As an application, we investigate an inverse source problem. Existence and uniqueness of the solution is proved. Moreover, we show that a solution pair is continuously depending on the initial and additional data, finalizing with a numerical test.

math.AP↗
arXiv2023

Inverse problem of determining time-dependent leading coefficient in the time-fractional heat equation

In this paper, we investigate direct and inverse problems for the time-fractional heat equation with a time-dependent leading coefficient for positive operators. First, we consider the direct problem, and the unique existence of the generalized solution is established. We also deduce some regularity results. Here, our proofs are based on the eigenfunction expansion method. Second, we study the inverse problem of determining the leading coefficient, and the well-posedness of this inverse problem is proved.

math.AP↗
arXiv2023

Heat Source Determining Inverse Problem for non-local in time equation

In this paper, we consider the inverse problem of determining the time-dependent source term in the general setting of Hilbert spaces and for general additional data. We prove the well-posedness of this inverse problem by reducing the problem to an operator equation for the source function.

math.AP↗
arXiv2023

Heat equation with singular thermal conductivity

In this paper, we study the heat equation with an irregular spatially dependent thermal conductivity coefficient. We prove that it has a solution in an appropriate very weak sense. Moreover, the uniqueness result and consistency with the classical solution if the latter exists are shown. Indeed, we allow the coefficient to be a distribution with a toy example of a Delta-function.

math.AP↗
arXiv2023

Schrödinger equation with singular position dependent mass

We consider the Schrödinger equation with singular position dependent effective mass and prove that it is very weakly well posed. A uniqueness result is proved in an appropriate sense, moreover, we prove the consistency with the classical theory. In particular, this allows one to consider Delta-like or more singular masses.

math.AP↗
arXiv2023

Some estimates for Mittag-Leffler function in quantum calculus and applications

The study of the Mittag-Leffler function and its various generalizations has become a very popular topic in mathematics and its applications. In the present paper we prove the following estimate for the $q$-Mittag-Leffler function: \begin{eqnarray*} \frac{1}{1+Γ_q\left(1-α\right)z}\leq e_{α,1}\left(-z;q\right)\leq\frac{1}{1+Γ_q\left(α+1\right)^{-1}z}. \end{eqnarray*} for all $0 < α< 1$ and $z>0$. Moreover, we apply it to investigate the solvability results for direct and inverse problems for time-fractional pseudo-parabolic equations in quantum calculus for a large class of positive operators with discrete spectrum.

math.AP↗
arXiv2023

Well-posedness of heat and wave equations generated by Rubin's q-difference operator in Sobolev spaces

In this paper, we investigate difference-differential operators of parabolic and hyperbolic types. Namely, we consider non-homogenous heat and wave equations for Rubin's difference operator. Well-posedness results are obtained in appropriate Sobolev type spaces. In particular, we prove that the heat and wave equations generated by Rubin's difference operator have unique solutions. We even show that these solutions can be represented by explicit formulas

math.AP↗
arXiv2021

The heat equation with singular potentials. II: Hypoelliptic case

In this paper we consider the heat equation with a strongly singular potential and show that it has a very weak solution. Our analysis is devoted to general hypoelliptic operators and is developed in the setting of graded Lie groups. The current work continues and extends a previous work , where the classical heat equation on $\mathbb R^n$ was considered.

math.AP↗
arXiv2021

Liouville theorems for Kirchhoff-type hypoelliptic Partial Differential Equations and systems. I. Heisenberg group

In this paper, we show the nonexistence results for the Kirchhoff elliptic, parabolic, and hyperbolic type equations on the Heisenberg groups. Also, the pseudo-parabolic and pseudo-hyperbolic equations of the Kirchhoff-type are under consideration. To prove these results we use the test function method. In addition, the analogous results are transferred to the cases of systems. Also, we give some examples of non-local nonlinearities.

math.AP↗
arXiv2021

Fractional Klein-Gordon equation with singular mass. II: Hypoelliptic case

In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution. Such analysis can be conveniently realised in the setting of graded Lie groups. The uniqueness of the very weak solution, and the consistency with the classical solution are also proved, under suitable considerations. This extends and improves the results obtained in the first part of this work which was devoted to the classical Euclidean Klein-Gordon equation.

math.AP↗
arXiv2021

Fractional Schrödinger Equation with singular potentials of higher-order

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 particle accumulating effect is observed in the singular cases. From the mathematical point of view, a "splitting of the strong singularity" phenomena is also observed.

math.AP↗