Search Results (86)

Several iterative integro-differential formulations for two-point, second- and third-order, nonlinear, boundary-value problems of ordinary differential equations based on Green’s functions and the method of variation of parameters are presented. It is shown that the generalized or dual Lagrange multiplier method (GVIM) previously developed for the iterative solution of nonlinear, boundary-value problems of ordinary differential equations that makes use of modified functionals and two Lagrange multipliers, is nothing but an iterative Green’s function formulation that does not require Lagrange multipliers at all. It is also shown that the two Lagrange multipliers of GVIM are associated with the left and right Green’s functions. The convergence of iterative methods based on both the Green function and the method of variation of parameters is proven for nonlinear functions that depend on the dependent variable and is illustrated by means of two examples. Several new iterative integro-differential formulations based on Green’s functions that use a multiplicative function for convergence acceleration are also presented. Full article

In this article, we investigate a nonlinear $\psi$-Hilfer fractional order Volterra integro-delay differential equation ($\psi$-Hilfer FRVIDDE) and a nonlinear $\psi$-Hilfer fractional Volterra delay integral equation ($\psi$-Hilfer FRVDIE), both of which incorporate multiple variable time delays. We establish sufficient conditions for the existence of a unique solution and the Ulam–Hyers stability (U-H stability) of both the $\psi$-Hilfer FRVIDDE and $\psi$-the Hilfer FRVDIE through two new main results. The proof technique relies on the Banach contraction mapping principle, properties of the Hilfer operator, and some additional analytical tools. The considered $\psi$-Hilfer FRVIDDE and $\psi$-Hilfer FRVDIE are new fractional mathematical models in the relevant literature. They extend and improve some available related fractional mathematical models from cases without delay to models incorporating multiple variable time delays, and they also provide new contributions to the qualitative theory of fractional delay differential and fractional delay integral equations. We also give two new examples to verify the applicability of main results of the article. Finally, the article presents substantial and novel results with new examples, contributing to the relevant literature. Full article

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 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 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 work introduces a new family of multivariate hybrid special polynomials, motivated by their growing relevance in mathematical modeling, physics, and engineering. We explore their core properties, including recurrence relations and shift operators, within a unified structural framework. By employing the factorization method, we derive various governing equations such as differential, partial differential, and integrodifferential equations. Additionally, we establish a related fractional Volterra integral equation, which broadens the theoretical foundation and potential applications of these polynomials. To support the theoretical development, we carry out computational simulations to approximate their roots and visualize the distribution of their zeros, offering practical insights into their analytical behavior. Full article

This work presents the mathematical frameworks of the “First-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Volterra-Type” (1st-FASAM-NIDE-V) and the “Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Volterra-Type” (2nd-FASAM-NIDE-V). It is shown that the 1st-FASAM-NIDE-V methodology enables the efficient computation of exactly-determined first-order sensitivities of the decoder response with respect to the optimized NIDE-V parameters, requiring a single “large-scale” computation for solving the 1st-Level Adjoint Sensitivity System (1st-LASS), regardless of the number of weights/parameters underlying the NIE-net. The 2nd-FASAM-NIDE-V methodology enables the computation, with unparalleled efficiency, of the second-order sensitivities of decoder responses with respect to the optimized/trained weights involved in the NIDE-V’s decoder, hidden layers, and encoder, requiring only as many “large-scale” computations as there are non-zero first-order sensitivities with respect to the feature functions. These characteristics of the 1st-FASAM-NIDE-V and 2nd-FASAM-NIDE-V are illustrated by considering a nonlinear heat conduction model that admits analytical solutions, enabling the exact verification of the expressions obtained for the first- and second-order sensitivities of NIDE-V decoder responses with respect to the model’s functions of parameters (weights) that characterize the heat conduction model. Full article

In this work, we address a nonlinear $\psi$-Hilfer fractional-order Volterra integro-differential equation that incorporates n-multiple-variable time delays. Employing the $\psi$-Hilfer fractional derivative operator, we investigate the existence of a unique solution, as well as the Ulam–Hyers–Rassias stability, semi-Ulam–Hyers–Rassias stability, and Ulam–Hyers stability of the proposed $\psi$-Hilfer fractional-order Volterra integro-differential equation through the fixed-point approach. In this study, we enhance and generalize existing results in the literature on $\psi$-Hilfer fractional-order Volterra integro-differential equations, both including and excluding single delay, by establishing new findings for nonlinear $\psi$-Hilfer fractional-order Volterra integro-differential equations involving n-multiple-variable time delays. This study provides novel theoretical insights that deepen the qualitative understanding of fractional calculus. Full article

The authors consider a nonlinear $\psi$-Hilfer fractional-order Volterra integro-differential equation ($\psi$-Hilfer FrOVIDE) that incorporates N-multiple variable time delays into the equation. By utilizing the $\psi$-Hilfer fractional derivative, they investigate the Ulam–Hyers–Rassias and Ulam–Hyers stability of the equation by using fixed-point methods. Their results improve existing ones both with and without delays by extending them to nonlinear $\psi$-Hilfer FrOVIDEs that incorporate N-multiple variable time delays. Full article

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 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 study investigates the stability behavior of nonlinear Fredholm and Volterra integral equations, as well as nonlinear integro-differential equations with Volterra integral terms, through the lens of symmetry principles in mathematical analysis. By leveraging fixed-point methods within b-metric spaces, which generalize classical metric spaces while preserving structural symmetry, we establish sufficient conditions for Hyers–Ulam–Rassias and Hyers–Ulam stability. The symmetric framework of b-metric spaces offers a unified approach to analyzing stability across a wide range of nonlinear systems. To illustrate the theoretical results, examples are provided that underscore the practical applicability and relevance of these findings to complex nonlinear systems, emphasizing their inherent symmetrical properties. Full article

The main aim of this study is to examine the Hyers–Ulam stability of fractional derivatives in Volterra–Fredholm integro-differential equations using Caputo fractional derivatives. We explore the existence and uniqueness of solutions for the proposed integro-differential equation using Banach and Krasnoselskii’s fixed-point techniques. Furthermore, we examine the Hyers–Ulam stability of the equation under the Caputo fractional derivative by deriving suitable sufficient conditions. We analyze the graphical behavior of the obtained results to demonstrate the efficiency of the analytical method, highlighting its ability to deliver accurate and precise approximate numerical solutions for fractional differential equations. Finally, numerical applications are presented to validate the stability of the proposed integro-differential equation. Full article

This paper presents an analysis of the existence, uniqueness, and stability of solutions to fractional neutral Volterra-Fredholm integro-differential equations, incorporating Caputo fractional derivatives and semigroup operators with state-dependent delays. By employing Krasnoselskii’s fixed point theorem, conditions under which solutions exist are established. To ensure uniqueness, the Banach Contraction Principle is applied, and the contraction condition is verified. Stability is analyzed using Ulam’s stability concept, emphasizing the resilience of solutions to perturbations and providing insights into their long-term behavior. An example is included, accompanied by graphical analysis that visualizes the solutions and their dynamic properties. Full article

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