Search Results (59)

The rapid densification of wireless networks demands efficient evaluation of special functions underpinning system-level performance metrics. To facilitate research, we introduce a computational framework tailored for the zero-balanced Gauss hypergeometric function $\Psi (x,y)\triangleq {}_{2}F_1(1,x;1+x;-y)$, a fundamental mathematical kernel emerging in Signal-to-Interference-plus-Noise Ratio (SINR) coverage analysis of non-uniform cellular deployments. Specifically, we propose a novel Reciprocal-Argument Transformation Algorithm (RTA), derived rigorously from a Mellin–Barnes reciprocal-argument identity, achieving geometric convergence with $O\left(1/y\right)$. By integrating RTA with a Pfaff-series solver into a hybrid algorithm guided by a golden-ratio switching criterion, our approach ensures optimal efficiency and numerical stability. Comprehensive validation demonstrates that the hybrid algorithm reliably attains machine-precision accuracy ($\sim {10}^{-16}$) within 1 μs per evaluation, dramatically accelerating calculations in realistic scenarios from hours to fractions of a second. Consequently, our method significantly enhances the feasibility of tractable optimization in ultra-dense non-uniform cellular networks, bridging the computational gap in large-scale wireless performance modeling. Full article

This paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional derivative operators, and $(k_2,{\psi }_2)$- Riemann–Liouville fractional integral operators. The problem considered in the present study is of a more general nature as the $(k_1,{\psi }_1)$-Hilfer fractional derivative operator specializes to several other fractional derivative operators by fixing the values of the function ${\psi }_1$ and the parameter $\beta$. Also the $(k_2,{\psi }_2)$-Riemann–Liouville fractional integral operator appearing in the multistrip boundary conditions is a generalized form of the ${\psi }_2$-Riemann–Liouville, $k_2$-Riemann–Liouville, and the usual Riemann–Liouville fractional integral operators (see the details in the paragraph after the formulation of the problem. Our study includes both convex and non-convex valued maps. In the upper semicontinuous case, we prove four existence results with the aid of the Leray–Schauder nonlinear alternative for multivalued maps, Mertelli’s fixed-point theorem, the nonlinear alternative for contractive maps, and Krasnoselskii’s multivalued fixed-point theorem when the multivalued map is convex-valued and $L^1$-Carathéodory. The lower semicontinuous case is discussed by making use of the nonlinear alternative of the Leray–Schauder type for single-valued maps together with Bressan and Colombo’s selection theorem for lower semicontinuous maps with decomposable values. Our final result for the Lipschitz case relies on the Covitz–Nadler fixed-point theorem for contractive multivalued maps. Examples are offered for illustrating the results presented in this study. Full article

Starting from ${\psi }_k$-Raina’s fractional integrals (${\psi }_k$-RFIs), the study obtains a new generalization of the Hermite–Hadamard–Mercer (H-H-M) inequality. Several trapezoid-type inequalities are constructed for functions whose derivatives of orders 1 and 2, in absolute value, are convex and involve ${\psi }_k$-RFIs. The results of the research are refinements of the Hermite–Hadamard (H-H) and H-H-M-type inequalities. For several types of fractional integrals—Riemann–Liouville (R-L), k-Riemann–Liouville (k-R-L), $\psi$-Riemann–Liouville ($\psi$-R-L), ${\psi }_k$-Riemann–Liouville (${\psi }_k$-R-L), Raina’s, k-Raina’s, and $\psi$-Raina’s fractional integrals ($\psi$-RFIs)—new inequalities of H-H and H-H-M-type are established, respectively. This article presents special cases of the main results and provides numerous examples with graphical illustrations to confirm the validity of the results. This study shows the efficiency of the findings with a couple of applications, taking into account the modified Bessel function and the q-digamma function. Full article

The primary aim of this manuscript is to establish unique fixed point results for a class of $\Psi$-contraction operators in complete G-metric spaces. By combining and extending various fixed point theorems in the context of $\Psi$-contraction operators, we introduce a novel function, denoted as $\psi$, and explore its properties. Our work presents new theoretical results, supported by examples and applications, that enrich the study of G-metric spaces. These results not only generalize and unify a broad range of existing findings in the literature but also expand their use to boundary value problems, Fredholm-type integral equations, and nonlinear Caputo fractional differential equations. In doing so, we offer a more comprehensive understanding of fixed point theory in the G-metric space framework and broaden its scope in applied mathematics. We also offer a numerical spectral approach for solving fractional initial value problems, utilizing shifted Chebyshev polynomials to construct a semi-analytic solution that inherently satisfies the given homogeneous initial conditions. Full article

The $(k,\psi )$-fractional derivative based on the k-gamma function is a more general version of the Hilfer fractional derivative. It is widely used in differential equations to describe physical phenomena, population dynamics, and biological genetic memory problems. In this article, we mainly study the $4m+2$-point symmetric integral boundary value problem of nonlinear $(k,\psi )$-fractional differential coupled Laplacian equations. The existence and uniqueness of solutions are obtained by the Krasnosel’skii fixed-point theorem and Banach’s contraction mapping principle. Furthermore, we also apply the calculus inequality techniques to discuss the stability of this system. Finally, three interesting examples and numerical simulations are given to further verify the correctness and effectiveness of the conclusions. Full article

In this work, we establish the existence of at least one solution for a p-Laplacian Langevin differential equation involving the $\psi$-Hilfer fractional derivative with antiperiodic boundary conditions. More precisely, we transform the studied problem into a Hammerstein integral equation, and after that, we use the Schafer fixed point theorem to prove the existence of at least one solution. Two examples are provided to validate the main result. Full article

Differential equations are frequently used to mathematically describe many problems in real life, but they are always subject to intrinsic phenomena that are neglected and could influence how the model behaves. In some cases like ecosystems, electrical circuits, or even economic models, the model may suddenly change due to outside influences. Occasionally, such changes start off impulsively and continue to exist for specific amounts of time. Non-instantaneous impulses are used in the creation of the models for this kind of scenario. In this paper, a new class of non-instantaneous impulsive $\psi$-Caputo fractional stochastic differential equations under integral boundary conditions driven by the Rosenblatt process was examined. Semigroup theory, stochastic theory, the Banach fixed-point theorem, and fractional calculus were applied to investigating the existence of piecewise continuous mild solutions for the systems under consideration. The impulsive Gronwall’s inequality was employed to establish the unique stability conditions for the system under consideration. Furthermore, we examined the controllability results of the proposed system. Finally, some examples were provided to demonstrate the validity of the presented work. Full article

In this article, we define a new class of noncommuting self mappings known as the S-operator pair. Also, we provide the existence and uniqueness of common fixed point results involving the S-operator pair satisfying the $(F,\phi ,\psi ,Z)$-contractive condition in m-metric spaces, which unifies and generalizes most of the existing relevant fixed point theorems. Furthermore, the variables in the m-metric space are symmetric, which is significant for solving nonlinear problems in operator theory. In addition, examples are provided in order to illustrate the concepts and results presented herein. It has been demonstrated that the results can be applied to prove the existence of a solution to a system of integral equations, a nonlinear fractional differential equation and an ordinary differential equation for damped forced oscillations. Also, in the end, the satellite web coupling problem is solved. Full article

The mathematical theories and methods of fractional calculus are relatively mature, which have been widely used in signal processing, control systems, nonlinear dynamics, financial models, etc. The studies of some basic theories of fractional differential equations can provide more understanding of mechanisms for the applications. In this paper, the expression of the Green function as well as its special properties are acquired and presented through theoretical analyses. Subsequently, on the basis of these properties of the Green function, the existence and uniqueness of positive solutions are achieved for a singular p-Laplacian fractional-order differential equation with nonlocal integral and infinite-point boundary value systems by using the method of a nonlinear alternative of Leray–Schauder-type Guo–Krasnoselskii’s fixed point theorem in cone, and the Banach fixed point theorem, respectively. Some existence results are obtained for the case in which the nonlinearity is allowed to be singular with regard to the time variable. Several examples are correspondingly provided to show the correctness and applicability of the obtained results, where nonlinear terms are controlled by the integrable functions ${\displaystyle \frac{1}{\pi {(lnt)}^{\frac{1}{2}}{(1-lnt)}^{\frac{1}{2}}}}$ and ${\displaystyle \frac{1}{\pi {(lnt)}^{\frac{3}{4}}{(1-lnt)}^{\frac{3}{4}}}}$ in Example 1, and by the integrable functions $\theta ,\overline{\theta }$ and $\phi \left(v\right),\psi \left(u\right)$ in Example 2, respectively. The present work may contribute to the improvement and application of the coupled p-Laplacian Hadamard fractional differential model and further promote the development of fractional differential equations and fractional differential calculus. Full article

This article aims to explore the existence and stability of solutions to differential equations involving a $\psi$-Hilfer fractional derivative in the Caputo sense, which, compared to classical $\psi$-Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the $\psi$-Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the $\psi$-Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a $\psi$-Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a $\psi$-Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies. Full article

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered $\Psi$–Caputo fractional derivative. These include equations with their right side depending on ordinary as well as fractional-order derivatives, or fractional integrals of the solution. Full article

In numerous scientific and engineering domains, fractional-order derivatives and integral operators are frequently used to represent many complex phenomena. They also have numerous practical applications in the area of fixed point iteration. In this article, we introduce the notion of generalized Meir-Keeler-Khan-Rational type $(\psi -\alpha )$-contraction mapping and propose fixed point results in partial metric spaces. Our proposed results extend, unify, and generalize existing findings in the literature. In regards to applicability, we provide evidence for the existence of a solution for the fractional-order differential operator. In addition, the solution of the integral equation and its uniqueness are also discussed. Finally, we conclude that our results are superior and generalized as compared to the existing ones. Full article

This research is concerned with the existence and uniqueness of solutions for a coupled system of $\Psi$–Riemann–Liouville fractional differential equations. To achieve this objective, we establish a set of necessary conditions by formulating the problem as an integral equation and utilizing well-known fixed-point theorems. By employing these mathematical tools, we demonstrate the existence and uniqueness of solutions for the proposed system. Additionally, to illustrate the practical implications of our findings, we provide several examples that showcase the main results obtained in this study. Full article

As claimed in many papers, the equivalence between the Caputo-type fractional differential problem and the corresponding integral forms may fail outside the spaces of absolutely continuous functions, even in Hölder spaces. To avoid such an equivalence problem, we define a “new” appropriate fractional integral operator, which is the right inverse of the Caputo derivative on some Hölder spaces of critical orders less than 1. A series of illustrative examples and counter-examples substantiate the necessity of our research. As an application, we use our method to discuss the BVP for the Langevin fractional differential equation $\frac{d_{\psi }^{\beta ,\mu }}{d{t}^{\beta }}\left(\frac{d_{\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}+\lambda \right)x\left(t\right)=f(t,x\left(t\right)),t\in [a,b],\lambda \in \mathbb{R}$, for $f\in C\left([a,b]\times \mathbb{R}\right)$ and some critical orders $\beta ,\alpha \in (0,1)$, combined with appropriate initial or boundary conditions, and with general classes of $\psi$-tempered Hilfer problems with $\psi$-tempered fractional derivatives. The BVP for fractional differential problems of the Bagley–Torvik type was also studied. Full article

In this paper, we introduce and thoroughly examine new generalized $\psi$-conformable fractional integral and derivative operators associated with the auxiliary function $\psi \left(t\right)$. We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit $\psi$-conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications. Full article