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12 pages, 269 KB

Open AccessArticle

Exponential Stability of Volterra Integro Dynamic Equations on Time Scales

by Andrejs Reinfelds and Shraddha Christian

Mathematics 2025, 13(24), 3918; https://doi.org/10.3390/math13243918 - 8 Dec 2025

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In this paper, we give new sufficient conditions for boundedness and exponential stability of solutions for nonlinear Volterra integro dynamic equations from above on unbounded time scales using first Lyapunovs method. To prove this result we reduce the n-dimensional problem to the [...] Read more.

In this paper, we give new sufficient conditions for boundedness and exponential stability of solutions for nonlinear Volterra integro dynamic equations from above on unbounded time scales using first Lyapunovs method. To prove this result we reduce the n-dimensional problem to the corresponding scalar one using the concept of matrix measure and a new simpler proof of Coppel’s inequality on the time scales. There is an example that illustrates the conditions of the theorem. Full article

11 pages, 622 KB

Open AccessArticle

Simple Two-Sided Convergence Method for a Special Boundary Value Problem with Retarded Argument

by Arzu Aykut, Ercan Çelik and İsrafil Okumuş

Axioms 2025, 14(12), 867; https://doi.org/10.3390/axioms14120867 - 26 Nov 2025

Viewed by 144

Abstract

This study utilizes approximation techniques to address a boundary value problem involving a differential equation with a delayed argument. The problem is approached through analytical techniques by transforming it firstly into an equivalent integral equation. Specifically, we derive a Fredholm–Volterra integral equation that [...] Read more.

This study utilizes approximation techniques to address a boundary value problem involving a differential equation with a delayed argument. The problem is approached through analytical techniques by transforming it firstly into an equivalent integral equation. Specifically, we derive a Fredholm–Volterra integral equation that encapsulates the delayed behavior inherent in the original differential equation. The Fredholm operator in this equation features a degenerate kernel, which enables simplification and facilitates the construction of successive approximations. To solve this integral equation, we employ the two-sided convergence method, a powerful iterative technique that generates two sequences of approximate solutions—lower and upper bounds—that converge monotonically toward the exact solution. This method is particularly well-suited for problems with delayed arguments, as it ensures both stability and convergence under appropriate conditions on the kernel functions. The main objective of the study is to demonstrate the applicability and accuracy of the Simple Two-Sided Convergence Method for this class of boundary value problems. A numerical example is presented to illustrate the theoretical results, and the obtained approximations are compared with the exact analytical solution. All computations were carried out using Maple, ensuring precise numerical evaluation. Full article

(This article belongs to the Special Issue Fractional Differentiation and Applied Mathematics: Recent Advances and Applications)

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21 pages, 1007 KB

Open AccessArticle

Ulam-Type Stability and Krasnosel’skii’s Fixed Point Approach for φ-Caputo Fractional Neutral Differential Equations with Iterated State-Dependent Delays

by Ravi P. Agarwal, Mihail M. Konstantinov and Ekaterina B. Madamlieva

Fractal Fract. 2025, 9(12), 753; https://doi.org/10.3390/fractalfract9120753 - 21 Nov 2025

Viewed by 537

Abstract

This work analyses the existence, uniqueness, and Ulam-type stability of neutral fractional functional differential equations with recursively defined state-dependent delays. Employing the Caputo fractional derivative of order

α \in (0, 1)

with respect to a strictly increasing function

φ

, [...] Read more.

This work analyses the existence, uniqueness, and Ulam-type stability of neutral fractional functional differential equations with recursively defined state-dependent delays. Employing the Caputo fractional derivative of order

α \in (0, 1)

with respect to a strictly increasing function

φ

, the analysis extends classical results to nonuniform memory. The neutral term and delay chain are defined recursively by the solution, with arbitrary continuous initial data. Existence and uniqueness of solutions are established using Krasnosel’skii’s fixed point theorem. Sufficient conditions for Ulam–Hyers stability are obtained via the Volterra-type integral form and a

φ

-fractional Grönwall inequality. Examples illustrate both standard and nonlinear time scales, including a Hopfield neural network with iterated delays, which has not been previously studied even for integer-order equations. Fractional neural networks with iterated state-dependent delays provide a new and effective model for the description of AI processes—particularly machine learning and pattern recognition—as well as for modelling the functioning of the human brain. Full article

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21 pages, 395 KB

Open AccessArticle

An Efficient Iteration Method for Fixed-Point Approximation and Its Application to Fractional Volterra–Fredholm Integro–Differential Equations

by Ekta Sharma, Shubham Kumar Mittal, Sunil Panday and Lorentz Jäntschi

Axioms 2025, 14(11), 830; https://doi.org/10.3390/axioms14110830 - 11 Nov 2025

Viewed by 443

Abstract

This paper proposes an efficient iteration method for fixed-point approximation in Banach spaces. The method accelerates convergence by incorporating a squared operator term within the iteration process. Analytical proofs verify its convergence and stability. Comparative numerical tests show that it converges faster and [...] Read more.

This paper proposes an efficient iteration method for fixed-point approximation in Banach spaces. The method accelerates convergence by incorporating a squared operator term within the iteration process. Analytical proofs verify its convergence and stability. Comparative numerical tests show that it converges faster and more reliably than established Picard-type methods. Its application to fractional models involving the Gamma function highlights the method’s efficiency and potential for broader use in nonlinear and fractional systems. Full article

(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)

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15 pages, 280 KB

Open AccessArticle

On Ćirić-Type Fixed Point Results on Interpolative b-Metric Spaces with Application to Volterra Integral Equations

by Pradip Debnath and Nabanita Konwar

Symmetry 2025, 17(11), 1914; https://doi.org/10.3390/sym17111914 - 8 Nov 2025

Viewed by 364

Abstract

This paper introduces a new class of generalized metric structures, called interpolative b-metric spaces, which unify and extend both b-metric spaces and interpolative metric spaces in a non-trivial way. By incorporating a nonlinear correction term alongside a multiplicative scaling parameter into [...] Read more.

This paper introduces a new class of generalized metric structures, called interpolative b-metric spaces, which unify and extend both b-metric spaces and interpolative metric spaces in a non-trivial way. By incorporating a nonlinear correction term alongside a multiplicative scaling parameter into the triangle inequality, this framework enables broader contractive conditions and refined control of convergence behavior. We develop the foundational theory of interpolative b-metric spaces and establish a generalized Ćirić-type fixed point theorem, along with Banach, Kannan, and Bianchini-type results as corollaries. To highlight the originality and applicability of our approach, we apply the main theorem to a nonlinear Volterra-type integral equation, demonstrating that interpolative b-metrics effectively accommodate nonlinear solution structures beyond the scope of traditional metric models. This work offers a unified platform for fixed point analysis and opens new directions in nonlinear and functional analysis. Full article

(This article belongs to the Topic Fixed Point Theory and Measure Theory)

19 pages, 1746 KB

Open AccessArticle

Coupled Multicomponent First Multiplicative Bogoyavlensky Lattice and Its Multisoliton Solutions

by Corina N. Babalic

Symmetry 2025, 17(11), 1907; https://doi.org/10.3390/sym17111907 - 7 Nov 2025

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Abstract

This study addresses the complete integrability of a generalized multicomponent version of the first multiplicative Bogoyavlensky lattice with branched dispersion. The analysis is performed using the Hirota bilinear formalism and the periodic reduction technique. Initially, a two-dimensional mB1 lattice is considered, for which [...] Read more.

This study addresses the complete integrability of a generalized multicomponent version of the first multiplicative Bogoyavlensky lattice with branched dispersion. The analysis is performed using the Hirota bilinear formalism and the periodic reduction technique. Initially, a two-dimensional mB1 lattice is considered, for which complete integrability is established by constructing its bilinear form and general multisoliton solutions via the Hirota bilinear formalism. A periodic reduction along the discrete independent variable is then applied to derive the coupled multicomponent mB1 lattice, along with its corresponding bilinear representation and multisoliton solutions. The resulting system serves as an integrable semi-discrete generalization of the classical Volterra-type equation. These findings contribute to the broader understanding of integrable lattice systems with branched dispersion relations and provide a constructive framework for obtaining explicit soliton solutions in multicomponent systems, which exhibit rich internal symmetry structures. Full article

(This article belongs to the Special Issue Symmetries in Applied and Theoretical Physics Selected Papers from the “12th BPU Congress”)

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12 pages, 673 KB

Open AccessArticle

Regularization of Nonlinear Volterra Integral Equations of the First Kind with Smooth Data

by Taalaibek Karakeev and Nagima Mustafayeva

AppliedMath 2025, 5(4), 146; https://doi.org/10.3390/appliedmath5040146 - 24 Oct 2025

Viewed by 346

Abstract

The paper investigates the regularization of solutions to nonlinear Volterra integral equations of the first kind, under the assumption that a solution exists and belongs to the space of continuous functions. The kernel of the equation is a differentiable function and vanishes on [...] Read more.

The paper investigates the regularization of solutions to nonlinear Volterra integral equations of the first kind, under the assumption that a solution exists and belongs to the space of continuous functions. The kernel of the equation is a differentiable function and vanishes on the diagonal at an interior point of the integration interval. By applying an appropriate differential operator (with respect to x), the Volterra integral equation of the first kind is reduced to a Volterra integral equation of the third kind, equivalent with respect to solvability. The subdomain method is employed by partitioning the integration interval into two subintervals. Within the imposed constraints, a compatibility condition for the solutions is satisfied at the junction point of the partial subintervals. A Lavrentiev-type regularizing operator is constructed that preserves the Volterra structure of the equation. The uniform convergence of the regularized solution to the exact solution is proved, and conditions ensuring the uniqueness of the solution in Hölder space are established. Full article

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11 pages, 263 KB

Open AccessArticle

Well-Posedness of Problems for the Heat Equation with a Fractional-Loaded Term and Memory

by Umida Baltaeva, Bobur Khasanov, Omongul Egamberganova and Hamrobek Hayitbayev

Dynamics 2025, 5(4), 44; https://doi.org/10.3390/dynamics5040044 - 14 Oct 2025

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Abstract

We investigate the Cauchy problem for a heat equation incorporating variable diffusion coefficients and fractional memory effects modeled by a separable convolution kernel. By employing the fundamental solution of the associated parabolic equation, the problem is reformulated as a Volterra-type integral equation. Under [...] Read more.

We investigate the Cauchy problem for a heat equation incorporating variable diffusion coefficients and fractional memory effects modeled by a separable convolution kernel. By employing the fundamental solution of the associated parabolic equation, the problem is reformulated as a Volterra-type integral equation. Under appropriate regularity assumptions, we establish existence and uniqueness of classical solutions. Furthermore, we address an inverse problem aimed at simultaneously recovering the memory kernel and the solution. Using a differentiability-based approach, we derive a stable and well-posed formulation that enables the identification of memory effects in fractional heat models. Full article

29 pages, 1831 KB

Open AccessArticle

On the Performance of Physics-Based Neural Networks for Symmetric and Asymmetric Domains: A Comparative Study and Hyperparameter Analysis

by Rafał Brociek, Mariusz Pleszczyński and Dawood Asghar Mughal

Symmetry 2025, 17(10), 1698; https://doi.org/10.3390/sym17101698 - 10 Oct 2025

Cited by 1 | Viewed by 808

Abstract

This work investigates the use of physics-informed neural networks (PINNs) for solving representative classes of differential and integro-differential equations, including the Burgers, Poisson, and Volterra equations. The examples presented are chosen to address both symmetric and asymmetric domains. PINNs integrate prior physical knowledge [...] Read more.

This work investigates the use of physics-informed neural networks (PINNs) for solving representative classes of differential and integro-differential equations, including the Burgers, Poisson, and Volterra equations. The examples presented are chosen to address both symmetric and asymmetric domains. PINNs integrate prior physical knowledge with the approximation capabilities of neural networks, allowing the modeling of physical phenomena without explicit domain discretization. In addition to evaluating accuracy against analytical solutions (where available) and established numerical methods, the study systematically examines the impact of key hyperparameters—such as the number of hidden layers, neurons per layer, and training points—on solution quality and stability. The impact of a symmetric domain on solution speed is also analyzed. The experimental results highlight the strengths and limitations of PINNs and provide practical guidelines for their effective application as an alternative or complement to traditional computational approaches. Full article

(This article belongs to the Special Issue Symmetry and Its Applications in Partial Differential Equations)

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16 pages, 374 KB

Open AccessArticle

An Extended Complex Method to Solve the Predator–Prey Model

by Hongqiang Tu and Guoqiang Dang

Axioms 2025, 14(10), 758; https://doi.org/10.3390/axioms14100758 - 10 Oct 2025

Viewed by 447

Abstract

Through transformation and utilizing a novel extended complex method combining with the Weierstrass factorization theorem, Wiman–Valiron theory and the Painlevé test, new non-constant meromorphic solutions were constructed for the predator–prey model. These meromorphic solutions contain the rational solutions, exponential solutions, elliptic solutions, and [...] Read more.

Through transformation and utilizing a novel extended complex method combining with the Weierstrass factorization theorem, Wiman–Valiron theory and the Painlevé test, new non-constant meromorphic solutions were constructed for the predator–prey model. These meromorphic solutions contain the rational solutions, exponential solutions, elliptic solutions, and transcendental entire function solutions of infinite order in the complex plane. The exact solutions contribute to understanding the predator–prey model from the perspective of complex differential equations. In fact, the presented synthesis method provides a new technology for studying some systems of partial differential equations. Full article

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20 pages, 769 KB

Open AccessArticle

Homotopy Analysis Method and Physics-Informed Neural Networks for Solving Volterra Integral Equations with Discontinuous Kernels

by Samad Noeiaghdam, Md Asadujjaman Miah and Sanda Micula

Axioms 2025, 14(10), 726; https://doi.org/10.3390/axioms14100726 - 25 Sep 2025

Viewed by 677

Abstract

This paper addresses first- and second-kind Volterra integral equations (VIEs) with discontinuous kernels. A hybrid method combining the Homotopy Analysis Method (HAM) and Physics-Informed Neural Networks (PINNs) is developed. The convergence of the HAM is analyzed. Benchmark examples confirm that the proposed HAM-PINNs [...] Read more.

This paper addresses first- and second-kind Volterra integral equations (VIEs) with discontinuous kernels. A hybrid method combining the Homotopy Analysis Method (HAM) and Physics-Informed Neural Networks (PINNs) is developed. The convergence of the HAM is analyzed. Benchmark examples confirm that the proposed HAM-PINNs approach achieves high accuracy and robustness, demonstrating its effectiveness for complex kernel structures. Full article

(This article belongs to the Special Issue Advances in Fixed Point Theory with Applications)

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16 pages, 1173 KB

Open AccessArticle

The Extended Goodwin Model and Wage–Labor Paradoxes Metric in South Africa

by Tichaona Chikore, Miglas Tumelo Makobe and Farai Nyabadza

Math. Comput. Appl. 2025, 30(5), 98; https://doi.org/10.3390/mca30050098 - 10 Sep 2025

Viewed by 752

Abstract

This study extends the post-Keynesian framework for cyclical economic growth, initially proposed by Goodwin in 1967, by integrating government intervention to stabilize employment amidst wage mismatches. Given the pressing challenges of unemployment and wage disparity in developing economies, particularly South Africa, this extension [...] Read more.

This study extends the post-Keynesian framework for cyclical economic growth, initially proposed by Goodwin in 1967, by integrating government intervention to stabilize employment amidst wage mismatches. Given the pressing challenges of unemployment and wage disparity in developing economies, particularly South Africa, this extension is necessary to better understand how policy interventions can influence labor market dynamics. Central to the study is the endogenous interaction between capital and labor, where class dynamics influence economic growth patterns. The research focuses on the competitive relationship between real wage growth and its effects on employment. Methodologically, the study measures the impact of intervention strategies using a system of coupled ordinary differential equations (Lotka–Volterra type), along with econometric techniques such as quantile regression, moving averages, and mean absolute error to measure wages mismatch. Results demonstrate the inherent contradictions of capitalism under intervention, confirming established theoretical perspectives. This work further contributes to economic optimality discussions, especially regarding the timing and persistence of economic cycles. The model provides a quantifiable approach for formulating intervention strategies to achieve long-term economic equilibrium and sustainable labor–capital coexistence. Full article

(This article belongs to the Section Natural Sciences)

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30 pages, 403 KB

Open AccessArticle

The Numerical Solution of Volterra Integral Equations

by Peter Junghanns

Axioms 2025, 14(9), 675; https://doi.org/10.3390/axioms14090675 - 1 Sep 2025

Viewed by 935

Abstract

Recently we studied a collocation–quadrature method in weighted

L^{2}

spaces as well as in the space of continuous functions for a Volterra-like integral equation of the form [...] Read more.

Recently we studied a collocation–quadrature method in weighted

L^{2}

spaces as well as in the space of continuous functions for a Volterra-like integral equation of the form

u (x) - \int_{α x}^{1} h (x - α y) u (y) d y = f (x), 0 < x < 1,

where

h (x)

(with a possible singularity at

x = 0

) and

f (x)

are given (in general complex-valued) functions, and

α \in (0, 1)

is a fixed parameter. Here, we want to investigate the same method for the case when

α = 1 .

More precisely, we consider (in general weakly singular) Volterra integral equations of the form

u (x) - \int_{0}^{x} h (x, y) {(x - y)}^{κ} u (y) d y = f (x), 0 < x < 1,

where

κ > - 1

, and

h : D ⟶ C

is a continuous function,

D = \{(x, y) \in R^{2} : 0 < y < x < 1\} .

The passage from

0 < α < 1

to

α = 1

and the consideration of more general kernel functions

h (x, y)

make the studies more involved. Moreover, we enhance the family of interpolation operators defining the approximating operators, and, finally, we ask if, in comparison to collocation–quadrature methods, the application of the Nyström method together with the theory of collectively compact operator sequences is possible. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

16 pages, 423 KB

Open AccessArticle

Numerical Solutions of Fractional Weakly Singular Two-Dimensional Partial Volterra Integral Equations Using Euler Wavelets

by Seyed Sadegh Gholami, Ali Ebadian, Amirahmad Khajehnasiri and Kareem T. Elgindy

Mathematics 2025, 13(17), 2718; https://doi.org/10.3390/math13172718 - 23 Aug 2025

Viewed by 720

Abstract

This paper presents an innovative numerical method for solving two-dimensional weakly singular Volterra integral equations, including fractional Volterra integral equations with weak singularities. Solving these equations in higher dimensions and in the presence of fractional and weak singularities is highly challenging. The proposed [...] Read more.

This paper presents an innovative numerical method for solving two-dimensional weakly singular Volterra integral equations, including fractional Volterra integral equations with weak singularities. Solving these equations in higher dimensions and in the presence of fractional and weak singularities is highly challenging. The proposed approach uses Euler wavelets (EWs) within an operational matrix (OM) framework combined with advanced numerical techniques, initially transforming these equations into a linear algebraic system and then solving it efficiently. This method offers very high accuracy, strong computational efficiency, and simplicity of implementation, making it suitable for a wide range of such complex problems, especially those requiring high speed and precision in the presence of intricate features. Full article

(This article belongs to the Special Issue Fractional Calculus: Advances and Applications)

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29 pages, 3058 KB

Open AccessArticle

Existence, Uniqueness, and Stability of Weighted Fuzzy Fractional Volterra–Fredholm Integro-Differential Equation

by Sahar Abbas, Abdul Ahad Abro, Syed Muhammad Daniyal, Hanaa A. Abdallah, Sadique Ahmad, Abdelhamied Ashraf Ateya and Noman Bin Zahid

Fractal Fract. 2025, 9(8), 540; https://doi.org/10.3390/fractalfract9080540 - 16 Aug 2025

Viewed by 893

Abstract

This paper investigates a novel class of weighted fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) subject to integral boundary conditions. The analysis is conducted within the framework of Caputo-weighted fractional calculus. Employing Banach’s and Krasnoselskii’s fixed-point theorems, we establish the existence and uniqueness of [...] Read more.

This paper investigates a novel class of weighted fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) subject to integral boundary conditions. The analysis is conducted within the framework of Caputo-weighted fractional calculus. Employing Banach’s and Krasnoselskii’s fixed-point theorems, we establish the existence and uniqueness of solutions. Stability is analyzed in the Ulam–Hyers (UHS), generalized Ulam–Hyers (GUHS), and Ulam–Hyers–Rassias (UHRS) senses. A modified Adomian decomposition method (MADM) is introduced to derive explicit solutions without linearization, preserving the problem’s original structure. The first numerical example validates the theoretical findings on existence, uniqueness, and stability, supplemented by graphical results obtained via the MADM. Further examples illustrate fuzzy solutions by varying the uncertainty level (r), the variable (x), and both parameters simultaneously. The numerical results align with the theoretical analysis, demonstrating the efficacy and applicability of the proposed method. Full article

(This article belongs to the Special Issue Recent Developments in Multidimensional Fractional Differential Equations and Fractional Difference Equations)

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