Mathematical Modelling and Analysis

A Vieta–Lucas collocation and non-standard finite difference technique for solving space-time fractional-order Fisher equation

Mohd Kashif — 2025-01-27

The purpose of the article is to analyze an accurate numerical technique to solve a space-time fractional-order Fisher equation in the Caputo sense. For this purpose, the spectral collocation technique is used, which is based on the Vieta–Lucas approximation. By using the properties of Vieta–Lucas polynomials, this technique reduces the nonlinear equations into a system of ordinary differential equations (ODEs). The non-standard finite difference (NSFD) method converts this system of ODEs into algebraic equations which have been solved numerically. Moreover, the error estimate is investigated for the proposed method. To show the accuracy and efficiency of the technique, the obtained numerical results are compared with the analytical results and existing results of the particular forms of the considered fractional order models through error analysis. The important feature of this article is the exhibition of variations of the field variable for various values of spatial and temporal fractional order parameters for different particular cases.

Dissipative measure-valued solutions to the magnetohydrodynamic equations

Jianwei Yang — 2025-01-27

In this paper, we study the dissipative measure-valued solution to the magnetohydrodynamic equations of 3D compressible isentropic flows with the adiabatic exponent γ > 1 and prove that a dissipative measure-valued solution is the same as the standard smooth classical solution as long as the latter exists, provided they emanate from the same initial data (weak–strong) uniqueness principle.

Using Chebyshev’s polynomials for solving Fredholm integral equations of the second kind

José Antonio Ezquerro — 2025-01-27

The main problem with the Newton method is the computation of the inverse of the first derivative of the operator involved at each iteration step. Thus, when we want to apply the Newton method directly to solve an integral equation, the existence of the inverse of the first derivative is guaranteed, when the kernel is sufficiently differentiable into any of its two components, through its approximation by Taylor’s polynomial. In this paper, we study the case in which the kernel is not differentiable in any of its two components. So, we present a strategy that consists of approximating the kernel of the nonlinear integral equation by a Chebyshev interpolation polynomial, which is separable. This allows us to explicitly calculate the inverse of the first derivative operator in each step of the Newton method and then approximate a solution of the approximate integral equation.

Solving class of mixed nonlinear multi-term fractional Volterra-Fredholm integro-differential equations by new development of HAM

Zainidin Eshkuvatov — 2025-01-27

This work implements the standard Homotopy Analysis Method (HAM) developed by Professor Shijun Liao (1992), and a new development of the HAM (called ND-HAM) improved by Z.K. Eshkuvatov (2022) in solving mixed nonlinear multi-term fractional derivative of different orders of Volterra-Fredholm Integrodifferential equations (FracVF-IDEs). Other than that, the existance and uniqueness of solution as well as the norm convergence with respect to ND-HAM, were proven in a Hilbert space. In addition, three numerical examples (including multi-term fractional IDEs) are presented and compared with the HAM, modified HAM and ”Generalized block pulse operational differentiation matrices method” developed in previous works by illustrating the accuracy as well as validity with respect to ND-HAM. Empirical investigations reveal that ND-HAM and the modified HAM yields the same results when control parameter ℏ is chosen as ℏ = −1 and is comparable to the standard HAM. The findings discovered that the ND-HAM is highly convenient, effective, as well as in line with theoretical results.

Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations

Zewen Wang — 2025-01-27

In this paper, we mainly study the numerical differentiation problem of computing the fractional order derivatives from noise data of a single variable function. Firstly, the numerical differentiation problem is reformulated into an inverse source problem of first order hyperbolic equation, and the corresponding solvability and the conditional stability are provided under suitable conditions. Then, four regularization methods are proposed to reconstruct the unknown source of hyperbolic equation which is the numerical derivative, and they are implemented by utilizing the finite dimensional expansion of source function and the superposition principle of hyperbolic equation. Finally, Numerical experiments are presented to show effectiveness of the proposed methods. It can be conclude that the proposed methods are very effective for small noise levels, and they are simpler and easier to be implemented than the previous PDEs-based numerical differentiation method based on direct and inverse problems of parabolic equations.

Discrete universality theorem for Matsumoto zeta-functions and nontrivial zeros of the Riemann zeta-function

Keita Nakai — 2025-01-27

In 2017, Garunkštis, Laurinčikas and Macaitienė proved the discrete universality theorem for the Riemann zeta-function shifted by imaginary parts of nontrivial zeros of the Riemann zeta-function. This discrete universality has been extended to various zeta-functions and L-functions. In this paper, we generalize this discrete universality for Matsumoto zeta-functions.

Nonstationary heat equation with nonlinear side condition

Tomas Belickas — 2025-01-27

The initial boundary value problem for the nonstationary heat equation is studied in a bounded domain with the specific overdetermination condition. This condition is nonlinear and can be interpreted as the energy functional. In present paper we construct the class of solutions to this problem.

Terminal value problem for the system of fractional differential equations with additional restrictions

Oleksandr Boichuk — 2025-01-23

This paper deals with the study of terminal value problem for the system of fractional differential equations with Caputo derivative. Additional conditions are imposed on the solutions of this problem in the form of a linear vector functional. Using the theory of pseudo-inverse matrices, we obtain the necessary and sufficient conditions for the solvability and the general form of the solution of this boundary-value problem. In the one-dimensional case, the obtained results are generalized to the case of a multi-point boundary-value problem. The issue of obtaining similar results for the terminal value problem for the system of fractional differential equations with tempered and Ψ–tempered fractional derivatives of Caputo type is considered.

Approximation of analytic functions by generalized shifts of the Lerch zeta-function

Aidas Balčiūnas — 2025-01-27

In the paper, we approximate analytic functions by generalized shifts of the Lerch zeta-function, where g is a certain increasing to real function having a monotonic derivative. We prove that, for arbitrary parameters λ and α, there exists a closed set of analytic functions defined in the strip 1/2 < σ < 1 which functions are approximated by the above shifts. If the set of logarithms is linearly independent over the field of rational numbers, then the set coincides with the set of all analytic functions in that strip.

A priori estimate and existence of solutions with symmetric derivatives for a third-order two-point boundary value problem

Sergey Smirnov — 2025-01-27

We study a priori estimate, existence, and uniqueness of solutions with symmetric derivatives for a third-order boundary value problem. The main tool in the proof of our existence result is Leray-Schauder continuation principle. Two examples are included to illustrate the applicability of the results.

Discrete Gronwall’s inequality for Ulam stability of delay fractional difference equations

Shu-Yu Yang — 2025-01-27

This paper investigates Ulam stability of delay fractional difference equations. First, a useful equality of double fractional sums is employed and discrete Gronwall’s inequality of delay type is provided. A delay discrete-time Mittag-Leffler function is used and its non-negativity condition is given. With the solutions’ existences, Ulam stability condition is presented to discuss the error estimation of exact and approximate solutions.