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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 124

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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25 pages, 10282 KB

Open AccessArticle

A Nonlinear Volterra Filtering Hybrid Image-Denoising Method Based on the Improved Bat Algorithm for Optimizing Kernel Parameters

by Wei Zhao, Chang-Bai Yu, Hai-Jun Liu and Yue Hu

Electronics 2025, 14(20), 4076; https://doi.org/10.3390/electronics14204076 - 16 Oct 2025

Viewed by 203

Abstract

To address the issue of reducing noise in images containing mixed noise, a Volterra filtering method based on a Bat algorithm with velocity weight perturbation is proposed to optimize kernel parameters. The structural advantages of the Volterra filter (predictive performance, linear and nonlinear [...] Read more.

To address the issue of reducing noise in images containing mixed noise, a Volterra filtering method based on a Bat algorithm with velocity weight perturbation is proposed to optimize kernel parameters. The structural advantages of the Volterra filter (predictive performance, linear and nonlinear terms) are used to reduce the noise in these images. The dynamic velocity inertia-weight perturbation mechanism is used to improve the Bat algorithm’s optimization ability, while the kernel-parameter optimization and the noise reduction abilities of the Volterra filter are further improved. Theoretical analysis and experimental results show that the high-density mixed noise, comprising Gaussian and salt-and-pepper noise, can be filtered effectively by the proposed algorithm. Compared to traditional image-denoising methods, the proposed method outperforms other algorithms in removing mixed noise from images while preserving edge details. Within a specific noise intensity range, the greater the intensity of mixed noise in the image, the better the noise reduction performance of this filtering method. The method proposed in this paper is less affected by noise intensity. When the number of bats in the population and the number of iterations reach a certain value, the algorithm exhibits good convergence and stability. Full article

(This article belongs to the Section Artificial Intelligence)

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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

Viewed by 287

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

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 415

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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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 552

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)

37 pages, 2744 KB

Open AccessArticle

Synergistic Evolution or Competitive Disruption? Analysing the Dynamic Interaction Between Digital and Real Economies in Henan, China, Based on Panel Data

by Yaping Zhu, Qingwei Xu, Chutong Hao, Shuaishuai Geng and Bingjun Li

Data 2025, 10(8), 126; https://doi.org/10.3390/data10080126 - 4 Aug 2025

Viewed by 829

Abstract

In the digital transformation era, understanding the relationship between digital and real economies is vital for regional development. This study analyses the interaction between these two economies in Henan Province using panel data from 18 cities (2011–2023). It incorporates policy support intensity through [...] Read more.

In the digital transformation era, understanding the relationship between digital and real economies is vital for regional development. This study analyses the interaction between these two economies in Henan Province using panel data from 18 cities (2011–2023). It incorporates policy support intensity through fuzzy set theory, applies an integrated weighting method to measure development levels, and uses regression models to assess the digital economy’s impact on the real economy. The coupling coordination degree model, kernel density estimation, and Gini coefficient reveal the coordination status and spatial distribution, while the ecological Lotka–Volterra model identifies the symbiotic patterns. The key findings are as follows: (1) The digital economy does not directly determine the state of the real economy. For example, cities such as Zhoukou and Zhumadian have low digital economy levels but high real economy levels. However, the development of the digital economy promotes the real economy without signs of diminishing returns. (2) The two economies are generally coordinated but differ spatially, with greater coordination in the Central Plains urban agglomeration. (3) The digital and real economies exhibit both collaboration and competition, with reciprocal mutualism as the dominant mode of integration. These insights provide guidance for policymakers and offer a new perspective on the integration of both economies. Full article

(This article belongs to the Special Issue Data in Behavioral and Experimental Research: Datasets and Applications)

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31 pages, 476 KB

Open AccessArticle

Strong Convergence of a Modified Euler—Maruyama Method for Mixed Stochastic Fractional Integro—Differential Equations with Local Lipschitz Coefficients

by Zhaoqiang Yang and Chenglong Xu

Fractal Fract. 2025, 9(5), 296; https://doi.org/10.3390/fractalfract9050296 - 1 May 2025

Viewed by 986

Abstract

This paper presents a modified Euler—Maruyama (EM) method for mixed stochastic fractional integro—differential equations (mSFIEs) with Caputo—type fractional derivatives whose coefficients satisfy local Lipschitz and linear growth conditions. First, we transform the mSFIEs into an equivalent mixed stochastic Volterra integral equations (mSVIEs) using [...] Read more.

This paper presents a modified Euler—Maruyama (EM) method for mixed stochastic fractional integro—differential equations (mSFIEs) with Caputo—type fractional derivatives whose coefficients satisfy local Lipschitz and linear growth conditions. First, we transform the mSFIEs into an equivalent mixed stochastic Volterra integral equations (mSVIEs) using a fractional calculus technique. Then, we establish the well—posedness of the analytical solutions of the mSVIEs. After that, a modified EM scheme is formulated to approximate the numerical solutions of the mSVIEs, and its strong convergence is proven based on local Lipschitz and linear growth conditions. Furthermore, we derive the modified EM scheme under the same conditions in the

L^{2}

sense, which is consistent with the strong convergence result of the corresponding EM scheme. Notably, the strong convergence order under local Lipschitz conditions is inherently lower than the corresponding order under global Lipschitz conditions. Finally, numerical experiments are presented to demonstrate that our approach not only circumvents the restrictive integrability conditions imposed by singular kernels, but also achieves a rigorous convergence order in the

L^{2}

sense. Full article

(This article belongs to the Section Numerical and Computational Methods)

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18 pages, 338 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Fundamental Matrix, Measure Resolvent Kernel and Stability Properties of Fractional Linear Delayed System with Discontinuous Initial Conditions

by Hristo Kiskinov, Mariyan Milev, Milena Petkova and Andrey Zahariev

Mathematics 2025, 13(9), 1408; https://doi.org/10.3390/math13091408 - 25 Apr 2025

Viewed by 448

Abstract

In the present work, a Cauchy (initial) problem for a fractional linear system with distributed delays and Caputo-type derivatives of incommensurate order is considered. As the main result, a new straightforward approach to study the considered initial problem via an equivalent Volterra–Stieltjes integral [...] Read more.

In the present work, a Cauchy (initial) problem for a fractional linear system with distributed delays and Caputo-type derivatives of incommensurate order is considered. As the main result, a new straightforward approach to study the considered initial problem via an equivalent Volterra–Stieltjes integral system is introduced. This approach is based on the existence and uniqueness of a global fundamental matrix for the corresponding homogeneous system, which allows us to prove that the corresponding resolvent system possesses a unique measure resolvent kernel. As a consequence, an integral representation of the solutions of the studied system is obtained. Then, using the obtained results, relations between the stability of the zero solution of the homogeneous system and different kinds of boundedness of its other solutions are established. Full article

(This article belongs to the Section C: Mathematical Analysis)

17 pages, 281 KB

Open AccessArticle

Fuzzy Double Yang Transform and Its Application to Fuzzy Parabolic Volterra Integro-Differential Equation

by Atanaska Georgieva, Slav I. Cholakov, Maria Vasileva and Yordanka Gudalova

Symmetry 2025, 17(4), 606; https://doi.org/10.3390/sym17040606 - 16 Apr 2025

Viewed by 571

Abstract

This article introduces a new fuzzy double integral transformation called fuzzy double Yang transformation. We review some of the main properties of the transformation and find the conditions for its existence. We prove the theorems for partial derivatives and fuzzy unitary convolution. All [...] Read more.

This article introduces a new fuzzy double integral transformation called fuzzy double Yang transformation. We review some of the main properties of the transformation and find the conditions for its existence. We prove the theorems for partial derivatives and fuzzy unitary convolution. All of the new results are applied to find an analytical solution to the fuzzy parabolic Volterra integro-differential equation (FPVIDE) with a suitably selected memory kernel. In addition, a numerical example is provided to illustrate how the proposed method might be helpful for solving FPVIDE utilizing symmetric triangular fuzzy numbers. Compared with other symmetric transforms, we conclude that our new approach is simpler and needs less calculations. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry in Fuzzy Control)

17 pages, 1187 KB

Open AccessArticle

Müntz–Legendre Wavelet Collocation Method for Solving Fractional Riccati Equation

by Fatemeh Soleyman and Iván Area

Axioms 2025, 14(3), 185; https://doi.org/10.3390/axioms14030185 - 2 Mar 2025

Cited by 2 | Viewed by 886

Abstract

We propose a wavelet collocation method for solving the fractional Riccati equation, using the Müntz–Legendre wavelet basis and its associated operational matrix of fractional integration. The fractional Riccati equation is first transformed into a Volterra integral equation with a weakly singular kernel. By [...] Read more.

We propose a wavelet collocation method for solving the fractional Riccati equation, using the Müntz–Legendre wavelet basis and its associated operational matrix of fractional integration. The fractional Riccati equation is first transformed into a Volterra integral equation with a weakly singular kernel. By employing the collocation method along with the operational matrix, we reduce the problem to a system of nonlinear algebraic equations, which is then solved using Newton–Raphson’s iterative procedure. The error estimate of the proposed method is analyzed, and numerical simulations are conducted to demonstrate its accuracy and efficiency. The obtained results are compared with existing approaches from the literature, highlighting the advantages of our method in terms of accuracy and computational performance. Full article

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

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19 pages, 798 KB

Open AccessArticle

Multifunctional Expectile Regression Estimation in Volterra Time Series: Application to Financial Risk Management

by Somayah Hussain Alkhaldi, Fatimah Alshahrani, Mohammed Kbiri Alaoui, Ali Laksaci and Mustapha Rachdi

Axioms 2025, 14(2), 147; https://doi.org/10.3390/axioms14020147 - 19 Feb 2025

Cited by 1 | Viewed by 1083

Abstract

We aim to analyze the dynamics of multiple financial assets with variable volatility. Instead of a standard analysis based on the Black–Scholes model, we proceed with the multidimensional Volterra model, which allows us to treat volatility as a stochastic process. Taking advantage of [...] Read more.

We aim to analyze the dynamics of multiple financial assets with variable volatility. Instead of a standard analysis based on the Black–Scholes model, we proceed with the multidimensional Volterra model, which allows us to treat volatility as a stochastic process. Taking advantage of the long memory function of this type of model, we analyze the reproduced movements using recent algorithms in the field of functional data analysis (FDA). In fact, we develop, in particular, new risk tools based on the asymmetric least squares loss function. We build an estimator using the multifunctional kernel (MK) method and then establish its asymptotic properties. The multidimensionality of the Volterra process is explored through the dispersion component of the convergence rate, while the nonparametric path of the risk tool affects the bias component. An empirical analysis is conducted to demonstrate the ease of implementation of our proposed approach. Additionally, an application on real data is presented to compare the effectiveness of expectile-based measures with Value at Risk (VaR) in financial risk management for multiple assets. Full article

(This article belongs to the Special Issue New Perspectives in Operator Theory and Functional Analysis)

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22 pages, 425 KB

Open AccessFeature PaperArticle

Extension of the First-Order Recursive Filters Method to Non-Linear Second-Kind Volterra Integral Equations

by Rodolphe Heyd

Mathematics 2024, 12(22), 3612; https://doi.org/10.3390/math12223612 - 19 Nov 2024

Cited by 1 | Viewed by 976

Abstract

A new numerical method for solving Volterra non-linear convolution integral equations (NLCVIEs) of the second kind is presented in this work. This new approach, named IIRFM-A, is based on the combined use of the Laplace transformation, a first-order decomposition, a bilinear transformation, and [...] Read more.

A new numerical method for solving Volterra non-linear convolution integral equations (NLCVIEs) of the second kind is presented in this work. This new approach, named IIRFM-A, is based on the combined use of the Laplace transformation, a first-order decomposition, a bilinear transformation, and the Adomian decomposition. Unlike most numerical methods based on the Laplace transformation, the IIRFM-A method has the dual advantage of requiring neither the calculation of the Laplace transform of the source function nor that of intermediate inverse Laplace transforms. The application of this new method to the case of non-convolutive multiplicative kernels is also introduced in this work. Several numerical examples are presented to illustrate the great flexibility and efficiency of this new approach. A concrete thermal problem, described by a non-linear convolutive Volterra integral equation, is also solved numerically using the new IIRFM-A method. In addition, this new approach extends for the first time the field of use of first-order recursive filters, usually restricted to the case of linear ordinary differential equations (ODEs) with constant coefficients, to the case of non-linear ODEs with variable coefficients. This extension represents a major step forward in the field of recursive filters. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

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12 pages, 397 KB

Open AccessArticle

Nonlocal Extensions of First Order Initial Value Problems

by Ravi Shankar

Axioms 2024, 13(8), 567; https://doi.org/10.3390/axioms13080567 - 21 Aug 2024

Viewed by 939

Abstract

We study certain Volterra integral equations that extend and recover first order ordinary differential equations (ODEs). We formulate the former equations from the latter by replacing classical derivatives with nonlocal integral operators with anti-symmetric kernels. Replacements of spatial derivatives have seen success in [...] Read more.

We study certain Volterra integral equations that extend and recover first order ordinary differential equations (ODEs). We formulate the former equations from the latter by replacing classical derivatives with nonlocal integral operators with anti-symmetric kernels. Replacements of spatial derivatives have seen success in fracture mechanics, diffusion, and image processing. In this paper, we consider nonlocal replacements of time derivatives which contain future data. To account for the nonlocal nature of the operators, we formulate initial “volume” problems (IVPs) for these integral equations; the initial data is prescribed on a time interval rather than at a single point. As a nonlocality parameter vanishes, we show that the solutions to these equations recover those of classical ODEs. We demonstrate this convergence with exact solutions of some simple IVPs. However, we find that the solutions of these nonlocal models exhibit several properties distinct from their classical counterparts. For example, the solutions exhibit discontinuities at periodic intervals. In addition, for some IVPs, a continuous initial profile develops a measure-valued singularity in finite time. At subsequent periodic intervals, these solutions develop increasingly higher order distributional singularities. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

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13 pages, 2982 KB

Open AccessArticle

Stability Analysis of the Solution for the Mixed Integral Equation with Symmetric Kernel in Position and Time with Its Applications

by Faizah M. Alharbi

Symmetry 2024, 16(8), 1048; https://doi.org/10.3390/sym16081048 - 14 Aug 2024

Viewed by 929

Abstract

Under certain assumptions, the existence of a unique solution of mixed integral equation (MIE) of the second type with a symmetric kernel is discussed, in

L_{2} [Ω] \times C [0, T], T < 1,

Ω

is the [...] Read more.

Under certain assumptions, the existence of a unique solution of mixed integral equation (MIE) of the second type with a symmetric kernel is discussed, in

L_{2} [Ω] \times C [0, T], T < 1,

Ω

is the position domain of integration and T is the time. The convergence error and the stability error are considered. Then, after using the separation technique, the MIE transforms into a system of Hammerstein integral equations (SHIEs) with time-varying coefficients. The nonlinear algebraic system (NAS) is obtained after using the degenerate method. New and special cases are derived from this work. Moreover, numerical results are computed using MATLAB R2023a software. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Equations: Mathematical Models, Methods and Applications Ⅱ)

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35 pages, 702 KB

Open AccessFeature PaperArticle

Numerical Solution of Linear Second-Kind Convolution Volterra Integral Equations Using the First-Order Recursive Filters Method

by Rodolphe Heyd

Mathematics 2024, 12(15), 2416; https://doi.org/10.3390/math12152416 - 3 Aug 2024

Cited by 1 | Viewed by 1495

Abstract

A new numerical method for solving Volterra linear convolution integral equations (CVIEs) of the second kind is presented in this work. This new approach uses first-order infinite impulse response digital filters method (IIRFM). Three convolutive kernels were analyzed, the unit kernel and two [...] Read more.

A new numerical method for solving Volterra linear convolution integral equations (CVIEs) of the second kind is presented in this work. This new approach uses first-order infinite impulse response digital filters method (IIRFM). Three convolutive kernels were analyzed, the unit kernel and two singular kernels: the logarithmic and generalized Abel kernels. The IIRFM is based on the combined use of the Laplace transformation, a first-order decomposition, and a bilinear transformation. This approach often leads to simple analytical expressions of the approximate solutions, enabling efficient numerical calculation, even using single-precision floating-point numbers. When compared with the method of homotopic perturbations with Laplace transformation (HPM-L), the IIRFM approach does not present, in linear cases, the convergence difficulties inherent to iterative approaches. Unlike most solution methods based on the Laplace transform, the IIRFM has the dual advantage of not requiring the calculation of the Laplace transform of the source function, and of not requiring the systematic calculation of inverse Laplace transforms. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

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