Search Results (45)

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

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order $\alpha$$\left(D^{\alpha }f\right)\left(y\right)=\frac{1}{\Gamma (m-\alpha )}{\int }_0^y{(y-x)}^{m-\alpha -1}{f}^{\left(m\right)}\left(x\right)dx,y>0$, with $m-1<\alpha \le m,m\in \mathbb{N}.$ The numerical procedure is based on approximating $f^{\left(m\right)}$ by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order $\alpha$. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure. Full article

The Chebyshev cardinal functions based on the Lobatto grid are introduced and used for the first time to solve the fractional delay differential equations. The presented algorithm is based on the collocation method, which is applied to solve the corresponding Volterra integral equation of the given equation. In the employed method, the derivative and fractional integral operators are expressed in the Chebyshev cardinal functions, which reduce the computational load. The method is characterized by its simplicity, adherence to boundary conditions, and high accuracy. An exact analysis has been provided to demonstrate the convergence of the scheme, and illustrative examples validate our investigation. Full article

The main objective of this work is to apply a novel and accurate algorithm for solving the second-order and fourth-order fractional diffusion-wave equations (FDWEs). First, the desired equation is reduced to the corresponding Volterra integral equation (VIE). Then, the collocation method is applied, for which the Chebyshev cardinal functions (CCFs) have been considered as the bases. In this paper, the CCFs based on a Lobatto grid are introduced and used for the first time to solve these kinds of equations. To this end, the derivative and fractional integral operators are represented in CCFs. The main features of the method are simplicity, compliance with boundary conditions, and good accuracy. An exact analysis to show the convergence of the scheme is presented, and illustrative examples confirm our investigation. Full article

This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to solving Volterra integral equations of the second kind and a fractional differential equation. The study focuses on a mixed domain where the parabolic section aligns with the upper half-plane, and the hyperbolic section is bounded by two characteristics of the equation under consideration and a segment of the x-axis. By utilizing the solution representation of the fractional-order diffusion equation, a primary functional relationship is derived between the traces of the sought function on the x-axis segment from the parabolic part of the mixed domain. An explicit solution form for the modified Cauchy problem in the hyperbolic section of the mixed domain is presented. This solution, combined with the problem’s boundary condition, yields a fundamental functional relationship between the traces of the unknown function, mapped to the interval of the equation’s degeneration line. Through the conjugation condition of the problem, an equation with fractional derivatives is obtained by eliminating one unknown function from two functional relationships. The solution to this equation is explicitly formulated. For a specific solution of the proposed problem, visualizations are provided for various orders of the fractional derivative. The analysis demonstrates that the derivative order influences both the intensity of the diffusion (or subdiffusion) process and the shape of the wave front. Full article

The extension of the theory of generalized fractal–fractional calculus, named in this article as Martínez–Kaabar Fractal–Fractional (MKFF) calculus, is addressed to the field of integral equations. Based on the classic Adomian decomposition method, by incorporating the MKFF $\alpha ,\gamma$-integral operator, we establish the so-called extended Adomian decomposition method (EADM). The convergence of this proposed technique is also discussed. Finally, some interesting Volterra Integral equations of non-integer order which possess a fractal effect are solved via our proposed approach. The results in this work provide a novel approach that can be employed in solving various problems in science and engineering, which can overcome the challenges of solving various equations, formulated via other classical fractional operators. Full article

The paper discusses the prospect of using a combined model based on finite segments (polynomials) of the Volterra integral power series. We consider a case when the problem of identifying the Volterra kernels is solved. The predictive properties of the classic Volterra polynomial are improved by adding a linear part in the form of an equivalent continued fraction. This technique allows us to distinguish an additional parameter—the connection coefficient $\alpha$, which is effective in adapting the constructed integral model to changes in technical parameters at the input of a dynamic system. In addition, this technique allows us to take into account the case of perturbing the kernel of the linear term of the Volterra polynomial in the metric $C[0,T]$ by a given value $\delta$, implying the ideas of Volterra regularizing procedures. The problem of choosing the connection coefficient is solved using a special extremal problem. The developed algorithms are used to solve the problem of identifying input signals of test dynamic systems, among which, in addition to mathematical ones, thermal power engineering devices are used. Full article

We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some numerical simulations demonstrate the efficiency and accuracy of the method. According to the obtained results, the presented method solves this type of equation well and gives significant results. Full article

In this paper, we derive the coincidence fixed-point and common fixed-point results for ℑ-type mappings satisfying certain contractive conditions and containing fewer conditions imposed on function ℑ with regard to generalized metric spaces (in terms of Jleli Samet). Finally, a fractional boundary value problem is reduced to an equivalent Volterra integral equation, and the existence results of common solutions are obtained with the use of proved fixed-point results. Full article