Search Results (38)

The Laplace equation is an important partial differential equation, typically used to describe the properties of steady-state distributions or passive fields in physical phenomena. Its Cauchy problem is one of the classic, serious, ill-posed problems, characterized by the fact that minor disturbances in the data can lead to significant errors in the solution and lack stability. Secondly, the determination of the parameters of the classical Laplace equation is difficult to adapt to the requirements of complex applications. For this purpose, in this paper, the Laplace equation with uncertain parameters is defined, and the uncertainty is represented by fuzzy numbers. In the case of granular differentiability, it is transformed into a granular differential equation, proving its serious ill-posedness. To overcome the ill-posedness, the Fourier regularization method is used to stabilize the numerical solution, and the stability estimation and error analysis between the regularization solution and the exact solution are given. Finally, numerical examples are given to illustrate the effectiveness and practicability of this method. Full article

The purpose of this paper is to investigate a class of novel symmetric coupled fuzzy fractional partial differential equation system involving the Caputo–Katugampola (C-K) generalized Hukuhara (gH) derivative. Within the framework of C-K gH differentiability, two types of gH weak solutions are defined, and their existence is rigorously established through explicit constructions via employing Schauder fixed point theorem, overcoming the limitations of traditional Lipschitz conditions and thereby extending applicability to non-smooth and nonlinear systems commonly encountered in practice. A typical numerical example with potential applications is proposed to verify the existence results of the solutions for the symmetric coupled system. Furthermore, we introduce Ulam–Hyers stability (U-HS) theory into the analysis of such symmetric coupled systems and establish explicit stability criteria. U-HS ensures the existence of approximate solutions close to the exact solution under small perturbations, and thereby guarantees the reliability and robustness of the systems, while it prevents significant deviations in system dynamics caused by minor disturbances. We not only enrich the theoretical framework of fuzzy fractional calculus by extending the class of solvable systems and supplementing stability analysis, but also provide a practical mathematical tool for investigating complex interconnected systems characterized by uncertainty, memory effects, and spatial dynamics. Full article

In this paper, we propose a fuzzy formulation of the classic Frankot–Chellappa algorithm by which surfaces can be reconstructed using normal vectors. In the fuzzy formulation, the surface normal vectors may be uncertain or ambiguous, yielding a fuzzy Poisson partial differential equation that requires appropriate definitions of fuzzy derivatives. The solution of the resulting fuzzy model is approached by adopting a fuzzy variant of the discrete sine transform, which results in a fast and robust algorithm for surface reconstruction. An adaptive defuzzification strategy is also introduced to improve noise handling in highly uncertain regions. In experiments, we demonstrate that our fuzzy Frankot–Chellappa algorithm achieves accuracy on par with the classic approach for smooth surfaces and offers improved robustness in the presence of noisy normal data. We also show that it can naturally handle missing data (such as gaps) in the normal field by filling them using neighboring information. Full article

Southwest China, with typical karst, is one of the 36 biodiversity hotspots in the world, facing extreme ecological fragility due to thin soils, limited water retention, and high bedrock exposure. This fragility intensifies under climate change and human pressures, threatening regional sustainable development. Ecological strategic areas (ESAs) are critical safeguards for ecosystem resilience, yet their spatiotemporal dynamics and driving mechanisms remain poorly quantified. To address this gap, this study constructed a multidimensional ecological health assessment framework (pattern integrity–process efficiency–function diversity). By integrating Sen’s slope, a correlated Mann–Kendall (CMK) test, the Hurst index, and fuzzy C-means clustering, we systematically evaluated ecological health trends and identified ESA differentiation patterns for 2000–2024. Orthogonal partial least squares structural equation modeling (OPLS-SEM) quantified driving factor intensities and pathways. The results revealed that ecological health improved overall but exhibited significant spatial disparity: persistently high in southern Guangdong and most of Yunnan, and persistently low in the Sichuan Basin and eastern Hubei, with 41.47% of counties showing declining/slightly declining trends. ESAs were concentrated in the southwest/southeast, whereas high-EHI ESAs increased while low-EHI ESAs declined. Additionally, the natural environmental and human interference impacts decreased, while unique geographic factors (notably the rock exposure rate, with persistently significant negative effects) increased. This long-term, multidimensional assessment provides a scientific foundation for targeted conservation and sustainable development strategies in fragile karst ecosystems. Full article

In this paper, we investigate a class of fuzzy partially differentiable models for Caputo–Hadamard-type Goursat problems with generalized Hukuhara difference, which have been widely recognized as having a significant role in simulating and analyzing various kinds of processes in engineering and physical sciences. By transforming the fuzzy partially differentiable models into equivalent integral equations and employing classical Banach and Schauder fixed-point theorems, we establish the existence and uniqueness of solutions for the fuzzy partially differentiable models. Furthermore, in order to overcome the complexity of obtaining exact solutions of systems involving Caputo–Hadamard fractional derivatives, we explore numerical approximations based on trapezoidal and Simpson’s rules and propose three numerical examples to visually illustrate the main results presented in this paper. Full article

In this paper, a novel approximate triangular fuzzy finite element method (FEM) is proposed to solve the one-dimensional second-order unsteady nonlinear fuzzy partial differential Boussinesq equation. The physical problem concerns the case of the drought flow of a horizontal unconfined aquifer with a limited breath B and special boundary conditions: (a) at x = 0, the water level is equal to zero, and (b) at x = B, the flow rate is equal to zero due to the presence of an impermeable wall. The initial water table is assumed to be curvilinear, following the form of an inverse incomplete beta function. To account for uncertainties in the system, the hydraulic parameters—hydraulic conductivity (K) and porosity (S)—are treated as fuzzy variables, considering sources of imprecision such as measurement errors and human-induced uncertainties. The performance of the proposed fuzzy FEM scheme is compared with the previously developed orthogonal fuzzy FEM solution as well as with an analytical solution. The results are in close agreement with those of the other methods, with the mean error of the analytical solution found to be equal to $1.19·{10}^{-6}$. Furthermore, the possibility theory is applied and fuzzy estimators constructed, leading to strong probabilistic interpretations. These findings provide valuable insights into the hydraulic properties of unconfined aquifers, aiding engineers and water resource managers in making informed and efficient decisions for sustainable hydrological and environmental planning. 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

In the real-world operation of unmanned helicopters, various state constraints, system uncertainties and multisource disturbances pose considerable risks to their safe fight. This paper focuses on anti-disturbance adaptive safety fixed-time control design for the uncertain unmanned helicopter subject to partial state constraints and multiple disturbances. Firstly, a developed safety protection algorithm is integrated with the fixed-time stability theory, which assures the tracking performance and guarantees that the partial states are always constrained within the time-varying safe range. Then, the compensation mechanism is developed to weaken the adverse impact induced by the filter errors. Simultaneously, the influence of the multisource disturbances on the system stability are weakened through the It$\widehat{o}$ differential equation and high-order disturbance observer. Further, the fuzzy logic system is constructed to approximate the system uncertainties caused by the sensor measurement errors and complex aerodynamic characteristics. Stability analysis proves that the controlled unmanned helicopter is semi-globally fixed-time stable in probability, and the state errors converge to a desired region of the origin. Finally, simulations are provided to illustrate the performance of the proposed scheme. Full article

The concept of summability is crucial in deriving formal solutions to partial differential equations. This paper explores the connection between the methods of statistical convergence of sequences and statistical Cesàro summability in intuitionistic fuzzy n-normed linear space (IFnNLS). While the existing literature covers Cesàro summability and its statistical variant in fuzzy, intuitionistic fuzzy, and classical normed spaces, this study stands out not only for its methodology but also for its comprehensive approach, encompassing a broader range of spaces and detailing the pathway from the statistical Cesàro summability method to statistical convergence. These results will lead us to Tauberian theorems in IFnNLS. Full article

Brazil has been one of the largest soybean producers in recent years. The soybean is a legume commonly found in family meals. Among the diseases affecting the grains, Asian soybean rust is one of the most concerning. The fungus causing the disease is spread by the wind, making it difficult to control. Although it has been researched since its first records, not much data are available regarding the macro propagation behavior of spores. Therefore, this research aimed to model its dispersion based on a partial differential equation, the diffusion–advection equation, used by researchers to model the behavior of any pollutant. The terms of this equation were developed from real data, processed by fuzzy logic, and the simulation results were compared with disease records throughout a harvest. By using this approach to model the spatiotemporal dynamics of this fungus, it was possible to simulate its spread satisfactorily. Additionally, its results were used as input variables for a fuzzy system that estimates the susceptibility of a given location to disease development. Full article

Flexible manipulators have been widely used in industrial production. However, due to the poor rigidity of the flexible manipulator, it is easy to generate vibration. This will reduce the working accuracy and service life of the flexible manipulator. It is necessary to suppress vibration during the operation of the flexible manipulator. Based on the energy method and the Hamilton principle, the partial differential equations of the manipulator were established. Secondly, an improved radial basis function (RBF) neural network was combined with the fuzzy backstepping method to identify and suppress random vibration during the operation of the flexible manipulator, and the Lyapunov function and control law were designed. Finally, Simulink was used to build a simulation platform, three different external disturbances were set up, and the effect of vibration suppression was observed through the change curves of the final velocity error and displacement error. Compared with the RBF neural network boundary control method and the RBF neural network inversion method, the simulation results show that the effect of the RBF neural network fuzzy inversion method is better than the previous two control methods, the system convergence is faster, and the equilibrium position error is smaller. Full article

In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo’s $gH$-differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo’s $gH$-differentiability pose challenges to dealing with Goursat problems. Therefore, in this study, we convert Goursat problems to equivalent systems fuzzy integral equations to deal properly with the mixed derivative term and two types of Caputo’s $gH$-differentiability. In this study, we utilize the concept of metric fixed point theory to discuss the existence of a unique solution of fractional fuzzy Goursat problems. For the useability of established theoretical work, we provide some numerical problems. We also discuss the solutions to numerical problems by conformable double Laplace transform. To show the validity of the solutions we provide 3D plots. We discuss, as an application, why fractional partial fuzzy differential equations are the generalization of usual partial fuzzy differential equations by providing a suitable reason. Moreover, we show the advantages of the proposed fractional transform over the usual Laplace transform. Full article

This article introduces an extension of classical fuzzy partial differential equations, known as fuzzy fractional partial differential equations. These equations provide a better explanation for certain phenomena. We focus on solving the fuzzy time diffusion equation with a fractional order of 0 < α ≤ 1, using two explicit compact finite difference schemes that are the compact forward time center space (CFTCS) and compact Saulyev’s scheme. The time fractional derivative uses the Caputo definition. The double-parametric form approach is used to transfer the governing equation from an uncertain to a crisp form. To ensure stability, we apply the von Neumann method to show that CFTCS is conditionally stable, while compact Saulyev’s is unconditionally stable. A numerical example is provided to demonstrate the practicality of our proposed schemes. Full article

In this study, we explore fractional partial differential equations as a more generalized version of classical partial differential equations. These fractional equations have shown promise in providing improved descriptions of certain phenomena under specific circumstances. The main focus of this paper comprises the development, analysis, and application of two explicit finite difference schemes to solve an initial boundary value problem involving a fuzzy time fractional convection–diffusion equation with a fractional order in the range of $0\le \xi \le 1$. The uniqueness of this problem lies in its consideration of fuzziness within both the initial and boundary conditions. To handle the uncertainty, we propose a computational mechanism based on the double parametric form of fuzzy numbers, effectively converting the problem from an uncertain format to a crisp one. To assess the stability of our proposed schemes, we employ the von Neumann method and find that they demonstrate unconditional stability. To illustrate the feasibility and practicality of our approach, we apply the developed scheme to a specific example. Full article

This study investigates the pricing formula for European options when the underlying asset follows a fuzzy mixed weighted fractional Brownian motion within a jump environment. We construct a pricing model for European options driven by fuzzy mixed weighted fractional Brownian motion with jumps. By converting the partial differential equation (PDE) into a Cauchy problem, we derive explicit solutions for both European call options and European put options. The figures and tables demonstrating the effectiveness of the results highlight the suitability of the fuzzy mixed weighted fractional Brownian motion with jump model for option pricing. Full article