Search Results (53)

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 introduces a fractional generalization of the classical Legendre differential equation based on the Atangana–Baleanu–Caputo (ABC) derivative. A novel fractional Legendre-type operator is rigorously defined within a functional framework of continuously differentiable functions with absolutely continuous derivatives. The associated initial value problem is reformulated as an equivalent Volterra integral equation, and existence and uniqueness of classical solutions are established via the Banach fixed-point theorem, supported by a proved Lipschitz estimate for the ABC derivative. A constructive solution representation is obtained through a Volterra–Neumann series, explicitly revealing the role of Mittag–Leffler functions. We prove that the fractional solutions converge uniformly to the classical Legendre polynomials as the fractional order approaches unity, with a quantitative convergence rate of order $O(1-\alpha )$ under mild regularity assumptions on the Volterra kernel. A fully reproducible quadrature-based numerical scheme is developed, with explicit kernel formulas and implementation algorithms provided in appendices. Numerical experiments for the quadratic Legendre mode confirm the theoretical convergence and illustrate the smooth interpolation between fractional and classical regimes. An application to time-fractional diffusion in spherical coordinates demonstrates that the operator arises naturally in physical models, providing a mathematically consistent tool for extending classical angular analysis to fractional settings with memory. Full article

This paper introduces a novel analytical and numerical framework for studying nonlinear fractional Volterra integral equations by employing an intuitionistic fuzzy metric structure over a Banach algebra. The principal contribution of this work is the development of fixed-point theory for a new class of intuitionistic fuzzy Z-contractions in $\mathcal{IFM}$-spaces over $\mathcal{BA}$, which extends existing fuzzy and algebra-valued metric frameworks. Within this setting, we established existence, uniqueness, and convergence results for solutions of fractional integral equations of the Caputo type by proving that the associated fractional integral operator satisfies the proposed contractive conditions. Furthermore, we demonstrated how the algebra-valued intuitionistic fuzzy structure enhances the analytical flexibility and robustness of the model. To support the theoretical findings, a numerical simulation based on a discretized iterative scheme is presented, illustrating the rapid convergence of the approximating sequence together with the monotone behavior of intuitionistic fuzzy nearness and non-nearness measures. The numerical results are consistent with the analytical theory and confirm the effectiveness of the proposed $\mathcal{IFM}$-spaces over the $\mathcal{BA}$ approach for fractional dynamical systems. Full article

This paper establishes a rigorous analytical framework for a nonlinear multi-order fractional differential system governed by the generalized $\sigma$-Hilfer operator in weighted Banach spaces. In contrast to existing studies that often treat specific kernels or fixed fractional orders in isolation, our approach provides a unified treatment that simultaneously handles multiple fractional orders, a tunable kernel $\sigma \left(\varsigma \right)$, weighted integral conditions, and a nonlinearity depending on a fractional integral of the solution. By converting the hierarchical differential structure into an equivalent Volterra integral equation, we derive sufficient conditions for the existence and uniqueness of solutions using the Banach contraction principle and Mönch’s fixed-point theorem with measures of non-compactness. The analysis is extended to Ulam–Hyers stability, ensuring robustness under modeling perturbations. A principal contribution is the systematic classification of the system’s symmetric reductions—specifically the Riemann–Liouville, Caputo, Hadamard, and Katugampola forms—all governed by a single spectral condition dependent on $\sigma \left(\varsigma \right)$. The theoretical results are illustrated by numerical examples that highlight the sensitivity of solutions to the memory kernel and the fractional orders. This work provides a cohesive analytical tool for a broad class of fractional systems with memory, thereby unifying previously disparate fractional calculi under a single, consistent framework. Full article

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 $\alpha \in (0,1)$ with respect to a strictly increasing function $\phi$, 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 $\phi$-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

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

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 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 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 study investigates extremal solutions for fractional-order delayed difference equations, utilizing the Caputo nabla operator to establish mild lower and upper approximations via discrete fractional calculus. A new approach is employed to demonstrate the uniform convergence of the sequences of lower and upper approximations within the monotone iterative scheme using the summation representation of the solutions, which serves as a discrete analogue to Volterra integral equations. This research highlights practical applications through numerical simulations in discrete bidirectional associative memory neural networks. Full article

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

In recent decades, many researchers have pointed out that derivatives and integrals of the non-integer order are well suited for describing various real-world materials, for example, polymers. It has also been shown that fractional-order mathematical models are more effective than integer-order mathematical models. Thereby, given these considerations, the investigation of qualitative properties, in particular, Ulam-type stabilities of fractional differential equations, fractional integral equations, etc., has now become a highly attractive subject for mathematicians, as this represents an important field of study due to their extensive applications in various branches of aerodynamics, biology, chemistry, the electrodynamics of complex media, polymer science, physics, rheology, and so on. Meanwhile, the qualitative concepts called Ulam–Hyers–Mittag-Leffler (U-H-M-L) stability and Ulam–Hyers–Mittag-Leffler–Rassias (U-H-M-L-R) stability are well-suited for describing the characteristics of fractional Ulam-type stabilities. The Banach contraction principle is a fundamental tool in nonlinear analysis, with numerous applications in operational equations, fractal theory, optimization theory, and various other fields. In this study, we consider a nonlinear fractional Volterra integral equation (FrVIE). The nonlinear terms in the FrVIE contain multiple variable delays. We prove the U-H-M-L stability and U-H-M-L-R stability of the FrVIE on a finite interval. Throughout this article, new sufficient conditions are obtained via six new results with regard to the U-H-M-L stability or the U-H-M-L-R stability of the FrVIE. The proofs depend on Banach’s fixed-point theorem, as well as the Chebyshev and Bielecki norms. In the particular case of the FrVIE, an example is delivered to illustrate U-H-M-L stability. Full article

This paper examines fractional multi-time scale stochastic functional differential equations that, in addition, are driven by fractional noises. Based on a specially crafted fixed-point principle for the so-called “local operators”, we prove a Peano-type theorem on the existence of weak solutions, that is, those defined on an extended stochastic basis. To encompass all commonly used particular classes of fractional multi-time scale stochastic models, including those with random delays and impulses at random times, we consider equations with nonlinear random Volterra operators rather than functions. Some crucial properties of the associated integral operators, needed for the proofs of the main results, are studied as well. To illustrate major findings, several existence theorems, generalizing those known in the literature, are offered, with the emphasis put on the most popular examples such as ordinary stochastic differential equations driven by fractional noises, fractional stochastic differential equations with variable delays and fractional stochastic neutral differential equations. Full article

This paper addresses the extension of Martinez–Kaabar (MK) fractal–fractional calculus (for simplicity, in this research work, it is referred to as MK calculus) to the field of integral transformations, with applications to some solutions to integral equations. A new notion of Laplace transformation, named MK Laplace transformation, is proposed, which incorporates the MK $\alpha ,\gamma$-integral operator into classical Laplace transformation. Laplace transformation is very applicable in mathematical physics problems, especially symmetrical problems in physics, which are frequently seen in quantum mechanics. Symmetrical systems and properties can be helpful in applications of Laplace transformations, which can help in providing an effective computational tool for solving such problems. The main properties and results of this transformation are discussed. In addition, the MK Laplace transformation method is constructed and applied to the non-integer-order first- and second-kind Volterra integral equations, which exhibit a fractal effect. Finally, the MK Abel integral equation’s solution is also investigated via this technique. Full article