Search Results (51)

In this study, we focus on solving the nonlinear time-fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) and modified Korteweg–de Vries (MKdV) equations, using the Yang transform iterative method (YTIM). This method combines the Yang transform with a new iterative scheme to construct reliable and efficient solutions. Readers can understand the procedures clearly, since the implementation of Yang transform directly transforms fractional derivative sections into algebraic terms in the given problems. The new iterative scheme is applied to generate series solutions for the provided problems. The fractional derivatives are considered in the Caputo sense. To validate the proposed approach, two numerical examples are analysed and compared with exact solutions, as well as with the results obtained from the fractional reduced differential transform method (FRDTM) and the q-homotopy analysis transform method (q-HATM). The comparisons, presented through both tables and graphical illustrations, confirm the enhanced accuracy and reliability of the proposed method. Moreover, the effect of varying the fractional order is explored, demonstrating convergence of the solution as the order approaches an integer value. Importantly, the time-fractional Hirota–Satsuma coupled KdV and modified Korteweg–de Vries (MKdV) equations investigated in this work are not only of theoretical and computational interest but also possess significant implications for achieving global sustainability goals. Specifically, these equations contribute to the Sustainable Development Goal (SDG) “Life Below Water” by offering advanced modelling capabilities for understanding wave propagation and ocean dynamics, thus supporting marine ecosystem research and management. It is also relevant to SDG “Climate Action” as it aids in the simulation of environmental phenomena crucial to climate change analysis and mitigation. Additionally, the development and application of innovative mathematical modelling techniques align with “Industry, Innovation, and Infrastructure” promoting advanced computational tools for use in ocean engineering, environmental monitoring, and other infrastructure-related domains. Therefore, the proposed method not only advances mathematical and numerical analysis but also fosters interdisciplinary contributions toward sustainable development. Full article

Research on the supply–demand relationships of ecosystem services (ESs) in alpine pastoral regions remains relatively scarce, yet it is crucial for regional ecological management and sustainable development. This study focuses on the Sanjiangyuan Region, a typical alpine pastoral area and significant ecological barrier, to quantitatively assess the supply–demand dynamics of key ESs and their spatial heterogeneity from 2000 to 2020. It further aims to elucidate the underlying driving mechanisms, thereby providing a scientific basis for optimizing regional ecological management. Four key ES indicators were selected: water yield (WY), grass yield (GY), soil conservation (SC), and habitat quality (HQ). ES supply and demand were quantified using an integrated approach incorporating the InVEST model, the Revised Universal Soil Loss Equation (RUSLE), and spatial analysis techniques. Building on this, the spatial patterns and temporal evolution characteristics of ES supply–demand relationships were analyzed. Subsequently, the Geographic Detector Model (GDM) and Geographically and Temporally Weighted Regression (GTWR) model were employed to identify key drivers influencing changes in the comprehensive ES supply–demand ratio. The results revealed the following: (1) Spatial Patterns: Overall ES supply capacity exhibited a spatial differentiation characterized by “higher values in the southeast and lower values in the northwest.” Areas of high ES demand were primarily concentrated in the densely populated eastern region. WY, SC, and HQ generally exhibited a surplus state, whereas GY showed supply falling short of demand in the densely populated eastern areas. (2) Temporal Dynamics: Between 2000 and 2020, the supply–demand ratios of WY and SC displayed a fluctuating downward trend. The HQ ratio remained relatively stable, while the GY ratio showed a significant and continuous upward trend, indicating positive outcomes from regional grass–livestock balance policies. (3) Driving Mechanisms: Climate and natural factors were the dominant drivers of changes in the ES supply–demand ratio. Analysis using the Geographical Detector’s q-statistic identified fractional vegetation cover (FVC, q = 0.72), annual precipitation (PR, q = 0.63), and human disturbance intensity (HD, q = 0.38) as the top three most influential factors. This study systematically reveals the spatial heterogeneity characteristics, dynamic evolution patterns, and core driving mechanisms of ES supply and demand in an alpine pastoral region, addressing a significant research gap. The findings not only provide a reference for ES supply–demand assessment in similar regions regarding indicator selection and methodology but also offer direct scientific support for precisely identifying priority areas for ecological conservation and restoration, optimizing grass–livestock balance management, and enhancing ecosystem sustainability within the Sanjiangyuan Region. Full article

This article presents the theory of trivariate q-truncated Gould–Hopper polynomials through a generating function approach utilizing q-calculus functions. These polynomials are subsequently examined within the framework of quasi-monomiality, leading to the establishment of fundamental operational identities. Operational representations are then derived, and q-differential and partial differential equations are formulated for the trivariate q-truncated Gould–Hopper polynomials. Summation formulae are presented to elucidate the analytical properties of these polynomials. Finally, graphical representations are provided to illustrate the behavior of trivariate q-truncated Gould–Hopper polynomials and their potential applications. Full article

This manuscript tries to show that there is only one solution to the problem of the q-Hilfer fractional generalized pantograph differential equations with a nonlocal condition, and it does so by employing a particular technique known as Schaefer’s fixed point theorem and the Banach contraction principle. Then, we verify that the Ulam-type stability is valid. To illustrate the results, an example is provided. Full article

This study investigates the fractional-order Sawada–Kotera and Rosenau–Hyman equations, which significantly model non-linear wave phenomena in various scientific fields. We employ two advanced methodologies to obtain analytical solutions: the q-homotopy Mohand transform method (q-HMTM) and the Mohand variational iteration method (MVIM). The fractional derivatives in the equations are expressed using the Caputo operator, which provides a flexible framework for analyzing the dynamics of fractional systems. By leveraging these methods, we derive diverse types of solutions, including hyperbolic, trigonometric, and rational forms, illustrating the effectiveness of the techniques in addressing complex fractional models. Numerical simulations and graphical representations are provided to validate the accuracy and applicability of derived solutions. Special attention is given to the influence of the fractional parameter on behavior of the solution behavior, highlighting its role in controlling diffusion and wave propagation. The findings underscore the potential of q-HMTM and MVIM as robust tools for solving non-linear fractional differential equations. They offer insights into their practical implications in fluid dynamics, wave mechanics, and other applications governed by fractional-order models. Full article

In the present investigation, we present certain subordination and superordination results for the q-integral operator of a fractional order associated with analytic functions in the open unit disk $\mathbf{U}$. Using this q-integral operator, we obtain sandwich-type results. Furthermore, we employ the existence of univalent solutions to a q-differential equation connected with a fractional q-integral operator of fractional order. We use these results to demonstrate the significance of our findings for particular functions. We also derive some examples and corollaries that are pertinent to our results. Full article

Computing the solution of the linear Caputo fractional differential equation with variable coefficients cannot be obtained in closed form as in the integer-order case. However, to use ‘q’, the order of the fractional derivative, as a parameter for our mathematical model, we need to compute the solution of the equation explicitly and/or numerically. The traditional methods, such as the integrating factor or variation of parameters methods used in the integer-order case, cannot be directly applied because the product rule of the integer derivative does not hold for the Caputo fractional derivative. In this work, we present a series solution method to compute the solution of the linear Caputo fractional differential equation with variable coefficients. This provides an opportunity to compare its solution with the corresponding integer solution, namely $q=1.$ Additionally, we develop a series solution method using analytic functions in the space of $C^q$ continuous functions. We also apply this series solution method to nonlinear Caputo fractional differential equations where the nonlinearity is in the form $f(t,u)=u^2.$ We have provided numerical examples to show the application of our series solution method. Full article

In the literature so far, for Caputo fractional boundary value problems of order $2q$ when $1<2q<2,$ the problems use the same boundary conditions of the integer-order differential equation of order ‘2’. In addition, they only use the left Caputo derivative in computing the solution of the Caputo boundary value problem of order $2q.$ Further, even the initial conditions for a Caputo fractional differential equation of order $nq$ use the corresponding integer-order initial conditions of order ‘n’. In this work, we establish that it is more appropriate to use the Caputo fractional initial conditions and Caputo fractional boundary conditions for sequential initial value problems and sequential boundary value problems, respectively. It is to be noted that the solution of a Caputo fractional initial value problem or Caputo fractional boundary value problem of order ‘$nq$’ will only be a $C^{nq}$ solution and not a $C^n$ solution on its interval. In this work, we present a methodology to compute the solutions of linear sequential Caputo fractional differential equations using initial and boundary conditions of fractional order $kq$, $k=0,1,\dots (n-1)$ when the order of the fractional derivative involved in the differential equation is $nq$. The Caputo left derivative can be computed only when the function can be expressed as $f(x-a)$. Then the Caputo right derivative of the same function will be computed for the function $f(b-x).$ Further, we establish that the relation between the Caputo left derivative and the Caputo right derivative is very essential for the study of Caputo fractional boundary value problems. We present a few numerical examples to justify that the Caputo left derivative and the Caputo right derivative are equal at any point on the Caputo function’s interval. The solution of the linear sequential Caputo fractional initial value problems and linear sequential Caputo fractional boundary value problems with fractional initial conditions and fractional boundary conditions reduces to the corresponding integer initial and boundary value problems, respectively, when $q=1.$ Thus, we can use the value of q as a parameter to enhance the mathematical model with realistic data. Full article

In the present study, we introduce the two-dimensional Chlodovsky-type Bernstein operators based on the $(p,q)$-integer. By leveraging the inherent symmetry properties of $(p,q)$-integers, we examine the approximation properties of our new operator with the help of a Korovkin-type theorem. Further, we present the local approximation properties and establish the rates of convergence utilizing the modulus of continuity and the Lipschitz-type maximal function. Additionally, a Voronovskaja-type theorem is provided for these operators. We also investigate the weighted approximation properties and estimate the rate of convergence in the same space. Finally, illustrative graphics generated with Maple demonstrate the convergence rate of these operators to certain functions. The optimization of approximation speeds by these symmetric operators during system control provides significant improvements in stability and performance. Consequently, the control and modeling of dynamic systems become more efficient and effective through these symmetry-oriented innovative methods. These advancements in the fields of modeling fractional differential equations and control theory offer substantial benefits to both modeling and optimization processes, expanding the range of applications within these areas. Full article

In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives and fractional derivatives in the Caputo sense to present the application of the method. The optimal q-HAM is an improved version of the Homotopy Analysis Method (HAM) and its modification q-HAM and focuses on finding the optimal value of the convergence parameters for a better approximation. Numerical applications are given where optimal values of the convergence control parameters are found. Additionally, the correspondence of the approximate solutions obtained for these optimal values and the exact or numerical solutions are shown with figures and tables. The results show that the optimal q-HAM improves the convergence of the approximate solutions obtained with the q-HAM. Approximate solutions obtained with the fractional Differential Transform Method, q-HAM and predictor–corrector method are also used to highlight the superiority of the optimal q-HAM. Analysis of the results from various methods points out that optimal q-HAM is a strong tool for the analysis of the approximate analytical solution in Abel-type differential equations. This approach can be used to analyze other fractional differential equations arising in mathematical investigations. Full article

This paper addresses the approximate controllability results for Hilfer fractional stochastic differential inclusions of order $1<\mathsf{q}<2$. Stochastic analysis, cosine families, fixed point theory, and fractional calculus provide the foundation of the main results. First, we explored the prospects of finding mild solutions for the Hilfer fractional stochastic differential equation. Subsequently, we determined that the specified system is approximately controllable. Finally, an example displays the theoretical application of the results. Full article

Computation of the solution of the nonlinear Caputo fractional differential equation is essential for using $q,$ which is the order of the derivative, as a parameter. The value of q can be determined to enhance the mathematical model in question using the data. The numerical methods available in the literature provide only the local existence of the solution. However, the interval of existence is known and guaranteed by the natural upper and lower solutions of the nonlinear differential equations. In this work, we develop monotone iterates, together with lower and upper solutions that converge uniformly, monotonically, and quadratically to the unique solution of the Caputo nonlinear fractional differential equation over its entire interval of existence. The nonlinear function is assumed to be the sum of convex and concave functions. The method is referred to as the generalized quasilinearization method. We provide a Caputo fractional logistic equation as an example whose interval of existence is $[0,\infty ).$ Full article

In this paper, we focus on a fractional differential equation ${\hspace{0.17em}}_0^C{D}^{\alpha }u\left(t\right)+q\left(t\right)u\left(t\right)=0$ with boundary value conditions $u\left(0\right)=\delta u\left(1\right),u^{\prime }\left(0\right)=\gamma u^{\prime }\left(1\right)$. The paper begins by pointing out the inadequacies of the study conducted by Ma and Yangin establishing Lyapunov-type inequalities. It then discusses the properties of its Green’s function and investigates extremum problems related to several linear functions. Finally, thorough classification and analysis of various cases for parameters $\delta$ and $\gamma$ are conducted. As a result, a comprehensive solution corresponding to the Lyapunov-type inequality is obtained. Full article

We investigate the transient dynamics of the Fisher equation under nonlinear diffusion and fractional operators. Firstly, we investigate the effects of the nonlinear diffusivity parameter in the integer-order Fisher equation, by considering a Gaussian distribution as the initial condition. Measuring the spread of the Gaussian distribution by $u{(0,t)}^{-2}$, our results show that the solution reaches a steady state governed by the parameters present in the logistic function in Fisher’s equation. The initial transient is an anomalous diffusion process, but a power law cannot describe the whole transient. In this sense, the main novelty of this work is to show that a q-exponential function gives a better description of the transient dynamics. In addition to this result, we extend the Fisher equation via non-integer operators. As a fractional definition, we employ the Caputo fractional derivative and use a discretized system for the numerical approach according to finite difference schemes. We consider the numerical solutions in three scenarios: fractional differential operators acting in time, space, and in both variables. Our results show that the time to reach the steady solution strongly depends on the fractional order of the differential operator, with more influence by the time operator. Our main finding shows that a generalized q-exponential, present in the Tsallis formalism, describes the transient dynamics. The adjustment parameters of the q-exponential depend on the fractional order, connecting the generalized thermostatistics with the anomalous relaxation promoted by the fractional operators in time and space. Full article

We introduce the concept of controlled extended Branciari quasi-b-metric spaces, as well as a ${\mathcal{G}}_q$-implicit type mapping. Under this new space setting, we derive some new fixed points, periodic points, right and left Ulam–Hyers stability, right and left weak well-posed properties, and right and left weak limit shadowing results. Additionally, we use these findings to solve the fractional differential equations of a Riesz–Caputo type with integral anti-periodic boundary values, as well of nonlinear matrix equations. All ideas, results, and applications are properly illustrated with examples. Full article