arXiv subjects

Arshyn Altybay

Publications and source records attributed to Arshyn Altybay.

12 recordsLinked to original sources

arXiv2026

Parabolic Equations with Singular Coefficients and Boundary Data: Analysis and Numerical Simulations

We investigate linear parabolic equations in divergence form with singular coefficients and nonsmooth initial--boundary data. When the diffusion, drift, or potential terms, as well as the source term and boundary conditions, are distributions rather than functions, classical and weak solution concepts become inadequate, since products involving distributions are not well defined in general. To address this difficulty, we introduce a framework of very weak solutions based on regularisation procedures and the theory of moderate nets. Under the stated moderateness and uniform-ellipticity assumptions, we establish existence of very weak solutions and prove uniqueness via negligibility arguments. Moreover, in the regular-data regime, we show consistency with classical weak solutions. Finally, we present numerical experiments illustrating the behaviour of regularised solutions for highly singular inputs, including delta-type potentials and distributional boundary traces.

math.AP↗

arXiv2026

Heat Equation driven by mixed local-nonlocal operators with non-regular space-dependent coefficients

In this paper, we study the Cauchy problem for a heat equation governed by a mixed local--nonlocal diffusion operator with spatially dependent coefficients. For bounded measurable coefficients satisfying uniform positivity conditions and initial data in $H^1(\mathbb R^d)$, we first establish existence and uniqueness of weak solutions in the natural energy space and derive an a priori energy estimate. We then extend the analysis to strictly positive distributional coefficients and compactly supported distributional initial data by means of a Friedrichs-type regularisation procedure. Under suitable moderateness assumptions on the regularising nets, we establish the existence of very weak solutions and prove uniqueness modulo negligible perturbations of admissible regularisations. Finally, under appropriate convergence assumptions on the regularised coefficients and initial data, together with preservation of uniform positivity, we prove consistency of the very weak solution with the corresponding weak solution.

math.AP↗

arXiv2026

Numerical identification of the time-dependent coefficient in the heat equation with fractional Laplacian

We consider the inverse problem of identifying a time-dependent source coefficient in a one-dimensional heat equation governed by the spectral Dirichlet fractional Laplacian from a weighted integral measurement. We present the continuous semigroup formulation and reduce the inverse problem to a Volterra equation under a nondegeneracy condition. For the numerical approximation, the spectral fractional operator is discretised by the fractional matrix power $A_h=L_h^s$ of the standard Dirichlet difference Laplacian and combined with a Crank--Nicolson scheme. We prove unconditional stability and an $O(τ^2+h^2)$ convergence estimate under a second-order spatial consistency assumption. A scalar reconstruction formula for the unknown coefficient is derived together with a discrete identifiability condition, and conditional convergence of the coupled state--coefficient reconstruction is established. For noisy integral measurements, regularised differentiation is used to approximate the required derivative data. Numerical experiments illustrate stable reconstruction for $1\%$--$5\%$ relative noise and are consistent with the derived conditional error estimate.

math.NA↗

arXiv2026

A backward problem for the time-fractional pseudo-parabolic equation with a variable coefficient

This work addresses an inverse reconstruction task for a time-fractional pseudo-parabolic model with a temporally varying coefficient. By imposing Dirichlet boundary conditions, we aim to recover the unknown initial state from observations collected at the final time. From a theoretical perspective, we derive existence and uniqueness results by proving that, under suitable hypotheses, the problem admits a unique solution. Computationally, we introduce a finite-difference discretisation based on a time-stepping strategy and provide a detailed stability and convergence analysis. Leveraging the resulting forward solver, we then formulate an initial-data identification procedure using Tikhonov regularisation. The proposed approach is validated with numerical simulations, and its resilience is assessed via experiments that incorporate perturbations in the final-time measurements.

math.NA↗

arXiv2026

Rothe's method in direct and time-dependent inverse source problems for a semilinear pseudo-parabolic equation

In this paper, we investigate the inverse problem of determining an unknown time-dependent source term in a semilinear pseudo-parabolic equation with variable coefficients and a Dirichlet boundary condition. The unknown source term is recovered from additional measurement data expressed as a weighted spatial average of the solution. By employing Rothe's time-discretisation method, we prove the existence and uniqueness of a weak solution under a smallness condition on the problem data. We also provide a numerical scheme based on a perturbation approach, which reduces the solution of the resulting discrete problem to solving two standard variational problems and evaluating a scalar coefficient, and we demonstrate its accuracy and stability through numerical experiments.

math.AP↗

arXiv2025

Numerical Approaches for Identifying the Time-Dependent Potential Coefficient in the Diffusion Equation

We address the inverse problem of identifying a time-dependent potential coefficient in a one-dimensional diffusion equation subject to Dirichlet boundary conditions and a nonlocal integral overdetermination constraint reflecting spatially averaged measurements. After establishing well-posedness for the forward problem and deriving an a priori estimate that ensures uniqueness and continuous dependence on the data, we prove existence and uniqueness for the inverse problem. To compute numerically the unknown coefficient, we propose and compare three numerical methods: an integration-based scheme, a Newton-Raphson iterative solver, and a physics-informed neural network (PINN). Numerical experiments on both exact and noisy data demonstrate the accuracy, robustness, and efficiency of each approach.

math.NA↗

arXiv2025

Numerical Identification of a Time-Dependent Coefficient in a Time-Fractional Diffusion Equation with Integral Constraints

In this paper, we numerically address the inverse problem of identifying a time-dependent coefficient in the time-fractional diffusion equation. An a priori estimate is established to ensure uniqueness and stability of the solution. A fully implicit finite-difference scheme is proposed and rigorously analysed for stability and convergence. An efficient algorithm based on an integral formulation is implemented and verified through numerical experiments, demonstrating accuracy and robustness under noisy data.

math.NA↗

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↗

arXiv2021

The heat equation with strongly singular potentials

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 physical observations.

math.AP↗

arXiv2021

Fractional Klein-Gordon equation with singular mass

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 $δ^2$ is observed.

math.AP↗

arXiv2020

A parallel hybrid implementation of the 2D acoustic wave equation

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 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 GPU, GPU+OpenMP, and Hybrid (GPU+OpenMP+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, CUDA, and CUDA+OpenMP implementations.

physics.comp-ph↗

arXiv2020

Tsunami propagation for singular topographies

We consider a tsunami wave equation with singular coefficients and prove that it has a very weak solution. Moreover, we show the uniqueness results and consistency theorem of the very weak solution with the classical one in some appropriate sense. Numerical experiments are done for the families of regularised problems in one- and two-dimensional cases. In particular, the appearance of a substantial second wave is observed, travelling in the opposite direction from the point/line of singularity. Its structure and strength are analysed numerically. In addition, for the two-dimensional tsunami wave equation, we develop GPU computing algorithms to reduce the computational cost.

math.AP↗