Results in Nonlinear Analysis

Transformation solution for Korteweg-de Vries equation with small delay

2025-03-03T13:20:12+03:00

In this paper we develop a new approach to get the transformation solution for the mathematical model of waves on shallow fluid; Korteweg-de Vries with a small delay without change the space variables. This method can be base to solve most of nonlinear higher order partial differential equation with time delay.

Bell Wavelet operational matrix method for convection diffusion equation

2025-02-06T11:35:05+03:00

In this article, we introduce an efficient method using Bell wavelets to solve fractional-order convection-diffusion equations with variable coefficients and initial boundary conditions. We begin by integrating block pulse functions with the Bell wavelet matrix to construct the fractional-order operational matrix of integration (OMI). This method simplifies fractional models by converting them into a set of algebraic equations via the collocation technique. The Bell wavelet collocation technique results in an efficient computational approach characterised by low costs and rapid convergence. Four numerical examples are presented, and the results are compared with exact solutions and other existing methods to validate the method and demonstrate its effectiveness and applicability. Graphical results highlight significant variations between fractional and integer orders, while our method adeptly handles both initial and boundary conditions, enhancing overall accuracy and simple applicability.

On Huang-Samet multivalued p-contractions

2025-03-03T14:12:06+03:00

This paper is devoted to prove the existence of fixed points for some classes of multivalued maps in the context of metric spaces. The obtained results generalize the recent theorems of Huang and Samet. Some examples are presented making effective our results.

Second Bounded Variation in the Sense of Shiba with Variable Exponent

2024-12-05T15:06:02+03:00

In this paper we present the notion of second bounded variation in the sense of Shiba with variable exponent, studying the structure of these functions spaces, showing its basic properties and some inclusion results among them.

Common fixed point theorems for compatible maps in complex valued b - metric space

2024-08-04T14:47:44+03:00

The article investigates common fixed point theorems in the framework of complete complex-valued b -metric space. It establishes such theorems for compatible mappings and employs rational inequalities to derive novel results. These findings serve to extend and generalize existing results in the literature. To illustrate the practical applicability and effectiveness of the proposed methods, the article presents several non-trivial examples.

A version of Hilbert's 13th problem for infinitely differentiable functions

2025-03-15T23:11:38+03:00

It is famous that Hilbert's 13th problem, asking if there exists a continuous real-valued function of multivariables which cannot be represented as any finite-time nested superposition of several functions of fewer variables, was proved by Kolmororov and Arnold. Actually, it is well known that there exist some other versions having been derived from the original one and still remaining to be open such as the analytic function version and the infinitely differentiable function version.

In this paper, we discuss a version of Hilbert's 13th problem for the infinitely differentiable functions. Exactly speaking, an example of an infinitely differentiable function of three real variables which cannot be represented as finite-time nested superposition of several infinitely differentiable functions of two real variables.

Eigen neutrosophic Z-set and neutrosophic Z-relation: A nonlinear dynamic modeling approach under uncertainty

2025-05-28T09:38:22+03:00

The modelling and Analysis of nonlinear systems under uncertainty are particularly difficult in scientific and engineering discipline. In many problems involving real world complex data, the inherent indeterminacy, vagueness and partial truth value associated with various real world complex phenomena is difficult to effectively capture using traditional frameworks. This study presents a novel nonlinear dynamic modelling framework based on Eigen Neutrosophic Z-Set and Neutrosophic Z-Relations framework to systematically handle the above challenges. The mathematical foundations of Neutrosophic Z sets are set up and discrete time dynamical systems given by Neutrosophic relational compositions are formulated. Properties of stability are rigorously analysed and proofs that
convergence to fixed point eigen structures are derived. Details are presented of computational algorithms for determining Greatest Eigen Neutrosophic Z-Set (GENZS) and Least Eigen Neutrosophic Z-Set (LENZS). The theoretical framework is validated through the simulations on representative systems in which the numerically generated solutions accommodate the fast convergence and retain
the stability of the systems under variation of the uncertainty conditions. The proposed approach provides a powerful and general means of modelling the nonlinear systems subject to uncertainty and could be applied, for example, in the area of engineering design, decision making systems and complex socio-economic modelling.

Influence of weather elements on grapes growth, yield and disease incidence using Epidemiological modeling

2024-07-23T13:59:30+03:00

Mathematical models play a vital role in crop agriculture to understand, predict and forecast the yield and disease incidence. This paper explores the utilization of mathematical model in grapes based on non-steady state equations possessing a nonlinear term. Theoretically, evaluated non-steady state concentration of leaf surface area for optimal strategy for grapes system presented. The analytical results of this study found to be matching with numerical results using MATLAB pdex4 function. Further, the proposed model validated with the field level data collected from Theni district, Tamil Nadu. Disease incidence and yield of grapes were highly influenced by weather variables. In addition, important aspects like disease intensity, infection rate and number of diseased leaves for secondary and primary infection studied in detail.

Approximating to a common fixed point and a minimizer of a convex function in Hadamard spaces

2025-04-30T14:33:10+03:00

This paper considers approximate sequences convergent to a common solution to a family of fixed point problems and convex minimization problems. We found that the lemma used to prove a known convergence theorem has a gap in its proof, and we obtained a counterexample. Further, we get an analogous result by substituting the convex combination of finitely many points by the balanced mapping.

Blow-Up Dynamics of Solutions to a Nonlinear Wave Equation with Positive Initial Energy

2025-02-04T13:48:29+03:00

This paper investigates the dynamics of a quasi-linear partial differential equation of fourth order characterized by bi-hyperbolic properties, incorporating dynamic boundary conditions. The study focuses on the interplay between the equation's nonlinear source term, the boundary effects, and the initial energy. By applying the concavity method, we derive conditions that lead to the finite-time blow-up phenomenon in solutions with non-negative initial energy. These findings highlight the impact of dynamic boundary conditions on the development of finite-time singularities in higher-order hyperbolic equations.

The Monto-Carlo procedure of some estimation methods for Exponential-Weibull distribution

2025-07-02T10:32:37+03:00

In this paper, deicing and derivative two parameters of Exponential Weibull (EW) distribution using three methods of estimate first is maximum likelihood estimation (MLE) the second is ordinary least squares estimation (OLS) and the third is ranked set sampling estimation (RSSE). After that, the simulation technique (Monto-Carlo method) is used to estimate the parameters and Reliability function for different values of sample size and various values of initial values by employing a MATLAB programmer and then using the Mean square error to find which method is the best

Mathematical modeling and optimization of intelligent systems using a hybrid PSO-GWO algorithm: A minx J(x) approach

2025-07-02T10:39:41+03:00

In this paper, a comparative analysis of Particle Swarm Optimization (PSO), Grey Wolf Optimization (GWO), and a hybrid PSO-GWO algorithm for the solution of complex optimization problems have been presented. The hybrid algorithm consists of the exploitation strength of GWO and the exploration capabilities of PSO while combining both together to surmount the failure of standalone algorithms like slow convergence in GWO and premature convergence in the PSO. The algorithms were evaluated in terms of convergence speed, robustness, and accuracy, on a series of benchmark functions (Sphere, Rastrigin, Ackley, Rosenbrock, and Griewank). The results of simulations indicate that the notion of a hybrid PSO-GWO algorithm is always better compared to standalone PSO as well as GWO for lower mean square errors (MSE) and quicker convergence to global optimum for different optimization landscapes. Finally, the hybrid approach took advantage of the best in all three aspects to optimize multimodal, non-convex, and deceptive functions with both reliable and robust performance that was superior to that of local minimum avoidance algorithms. This research shows that the hybrid PSO-GWO algorithm is an efficient tool for robust optimization in the real-world systems of dynamic, complex systems. Future work is to extend the algorithm’s adaptability to real-world constraints, and dynamic parameter adjustment, and integrate it within domain-specific heuristics to improve the optimization in engineering and automation tasks.

Optimization of ordinary differential equations (ODE) solutions using modified recurrent neural networks

2025-07-02T10:51:57+03:00

This paper introduces a hybrid approach for solving ordinary differential equations (ODE) using modified recurrent neural networks (mRNNs). The approach combines mRNNs with novel optimization techniques. Crucially, when training an mRNN, training data points should be selected from the open interval (a, b) to avoid training the network with the boundary points. This approach reduces computational errors by avoiding boundary region training. Furthermore, we propose a transformation that maps training points from a potentially broader interval [a, b] into corresponding points within the open interval (a, b), before training. This allows the network to be trained on points that are similar in the open interval, which leads to improved accuracy. The proposed model demonstrates higher
accuracy compared to existing mRNN models. A numerical example and corresponding simulations demonstrate the mathematical effectiveness of this approach.

Some Novel Versions of Fractional Hermite–Hadamard-Mercer Type Inequalities with Matrix Applications

2025-04-17T09:57:21+03:00

In this study, we explore several fractional Hermite–Hadamard (H–H)-Mercer inequalities for interval-valued functions through the use of a generalized fractional integral operator (GFIO). Furthermore, we examine new variations of the H–H-Mercer inequality in relation to GFIO. Various examples are included to support our assertions. The results could offer new insights into a broad spectrum of integral inequalities for fractional fuzzy systems within the framework of interval analysis, along with the optimization issues they raise. Moreover, some applications on matrices are illustrated.

An extended iterative algorithm for solving nonlinear systems and differential equations

2025-07-17T10:31:51+03:00

An extended iterative algorithm for solving nonlinear systems and ordinary differential equations (ODE) is presented in this paper which is efficient. Convergence difficulty, high computation cost and instability are the obstacles of using traditional methods like Newton method and explicit Runge-Kutta methods for stiff problems. In this approach proposed, we enhance numerical stability and bring about the convergence speed by introducing a modifed Jacobian matrix. To solve multi variable nonlinear systems, the algorithm is structured such that it adapts to a dynamically varied update step, which reduces the sensitivity toward the initial conditions. Furthermore, for ODEs we apply an implicit numerical integration combining with the modified Jacobian so that it is well suited to stiff and high dimensional systems. It is shown by theoretical analysis that the nonlinear equations converge superlinearly and numerical experiments are used to show that the methods perform better than classical approaches in terms of accuracy and efficiency. It is used for engineering simulations as well as machine learning optimization problems. The method will be extended to partial differential equations (PDEs) as future work, as will adaptively choosing the step size for further improvements in the computation

Utilizing various statistical techniques to estimate the scale parameter for the Rayleigh distribution

2025-06-12T09:53:35+03:00

The purpose of this paper is to use statistical estimating techniques to estimate the scale parameter of the Rayleigh distribution (maximum likelihood, rank set sampling, Bayes, and robust Bayes estimators) under complete data. A non-Bayes estimator is obtained by rank set sampling and maximum likelihood. The inverted gamma distribution and the inverted exponential distribution based on symmetric unbalanced (squared error) and balanced (squared) loss functions were used as Bayesian estimators. Robust Bayes analysis for the Rayleigh distribution depends on (unbalanced and balanced) loss functions based on (ML-II-ϵ-contaminated class and derived under the prior contaminant distribution of the Frechet distribution. The estimators' performances were contrasted according to simulation experiments for different cases and sample sizes depending on the value for the mean squared error.

Solving fuzzy system for Volterra-Fredholm integral equations of the second kind using homotopy perturbation method

2025-06-20T10:36:33+03:00

In the current research, the linear fuzzy system of the Volterra-Fredholm integral equations (FSVFIEs) is solved using the homotopy perturbation method (HPM). The Banach contraction fixed point is used to demonstrate the convergence of the solution under the established approximate scheme. The symmetric fuzzification solution is obtained by using convex symmetrical triangular fuzzy numbers. To verify the accuracy and efficacy of this method to handling FSVFIEs, the approximate and exact solutions are compared. Results from numerically solving examples of FSVFIEs are used to test the effectiveness of the suggested approach. These results show that the suggested strategy is highly effective and that the suggested method is easy to use.

Fixed point theorems in fuzzy modular metric spaces for non-expansive mappings with applications to neural operator stability

2025-07-26T18:53:29+03:00

It is also the aim of the paper to introduce new fixed-point results in the context of fuzzy modular metric spaces in connection to non-expansive maps and how they can be applied to the stability of neural operators. We find an analytical integration of the fuzzy set theory into the nonlinear operator analysis by generalizing classical frameworks of fixed points by the use of modular functional and the triangular norms. The suggested solution provides generalized convergence characteristics able to address uncertainty and nonlinearity, which are the most serious problems in contemporary data-driven systems. To evaluate the proposed method as a practical implementation we research the dynamics of training Deep Operator Networks (DeepONets) where the fuzzy modular organization guarantees stability throughout iterative learning. Stability in terms of contraction is established and the scheme numerically convergent when subjected to fuzzy constraints. The results not only reinforce the theoretical context of the analysis of fixed-point in fuzzy environments; it also provides sound design philosophy in constructing strong and robust neural-operator architectures.

Fuzzy logic-driven nonlinear optimization for resource allocation in 6G networks

2025-08-02T06:55:24+03:00

Since the last-generation (6G) wireless networks strive to sustain ultra-dense, ultra-low-latency, and ultra-high-diversity systems, an essential challenge is how to manage resource scheduling in a realtime under uncertainty. Conventional deterministic optimization techniques hit their limit when they are confronted by the factors of non-stationary channel operation, dynamic user mobility, non predictable quality-of-service (QoS) needs, and distributed network control. Using fuzzy logic to drive nonlinear optimization of resource allocation in 6G networks is a proposed novel framework displayed in the paper. The suggested model incorporates fuzzy sets and nonlinear principles of optimality in humanlike reasoning and ambiguity to reshape policies of resources distributions in complicated wireless settings. The resource allocation problem is modeled as a fuzzy-constrained nonlinear program in which delay, QoS satisfaction and channel stability are considered as fuzzy parameters with membership functions related to them. The major contributions can be summarized as four: (1), development of a fuzzy module to model soft constraint of latency and throughput, (2), a nonlinear utility maximization resource allocation function which involves scalable service-level defuzzification, (3), convergence and existence of a solution as proved through the fuzzy variational inequality, (4), simulation and analysis of soft-constrained fuzzy optimization under numerous conditions of fading, mobility and user load. Python simulations on ISO/ITU-standard 6G use cases present significant performance gains with respect to conventional convex and heuristic plans: more than 27.4 improvement in spectral use, 18.2 improvement in packet latency violation, and 21.5 improvement on user fairness. Moreover, the model is robust, both in situations of partial observability, and in conditions of asymmetric information and the model works well under uncertainty in specifying drop rates and retransmission overhead. This collaborative way of working the sides of fuzzy reasoning structures, nonlinear optimization solver, and exploits of models of future 6 G systems offers a mathematically sound and practically competent design of adaptive intelligent wireless networks. The paper is a foundation in the future research on fuzzy cooperative game theory, hierarchically partitioned RIS-aided systems and the use of learning aids in fuzzy controllers to optimize wireless systems in the light of uncertainty.

Higher-order conformable derivatives and their applications

2025-07-25T09:43:34+03:00

After presenting some results and introducing a new definition of the higher-order conformable derivative, we discuss how these findings relate to a novel idea: The higher-order conformable Laplace transform and the higher-order conformable Sumudu transform.

Characterizing two inclusive subfamilies of complex order defined by error functions and subordinate to horadam polynomials

2025-06-12T10:53:08+03:00

In this paper, using the Error functions and subordinate to Horadam polynomials, we introduce two inclusive subfamilies AEH( , , , , , ) and BEH( , , , , , ) of complex order. For functions in these subfamilies, we derive the estimations of the initial coefficients | | Q2 and | | Q3 , as well as the Fekete-Szegö functional. Further, some related results are also obtained as corollaries and remark.

Investigation of near fixed points, near fixed interval ellipse and its equivalence classes

2025-07-25T09:48:03+03:00

The objective of the manuscript is to employ the Hardy-Roger contraction to determine the near fixed point and its unique equivalence class in the context of the b −interval metric space. Further, an improved b −interval metric variant of a quasi-contraction characterizing the completeness of a b − interval metric space is exhibited. Various illustrations have been provided to show the existence of a near fixed point and its distinct equivalence class for both continuous and discontinuous maps developed in the b −interval metric space. As an application of the b −interval metric, a near-fixed interval ellipse and its unique equivalence  −class are introduced to study the geometry of non-unique near fixed points.

Perturbed statistical convergence

2025-07-30T13:18:05+03:00

This paper examines the basic features of perturbed statistical convergence in the context of perturbed metric spaces. The suggested method expands on the standard concept of statistical convergence by using a perturbation function that shows the errors that might happen while measuring distance. The relations of this new type of convergence with classical and statistical convergence are discussed in detail. There are some examples and counterexamples that support the new theoretical results.

Nonlinear optimization-driven deep learning framework for medical image reconstruction via partial differential equations

2025-08-29T16:22:22+03:00

High quality medical imaging is essential to accurate clinical decision-making, but reconstruction of sparse or noisy images especially under CT and MRI is still a major challenge, with traditional reconstruction algorithms vulnerable to artifacts and noise, and unwanted inference typically lacking interpretability. We introduce a new modality of addressing the problem of reconstruction with non-linear optimization, partial differential equation (PDE) constraints and deep neural networks, where the priors on physical properties should be presented as the network loss function and the architecture of the network so as to build more robust and accurate reconstruction. Having a clear formulation of a nonlinear optimization problem and by using the principles of variational approaches, we are
also able to integrate a hardware friendly circuit into our solution that could be used to acquire data in real time. Experiments on benchmark CT and MRI, indicate an increase in peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), more effective noise suppression, and convergence times than state-of-the art baselines. The need of nonlinear optimization and employment of PDEs regularization in work on edges preservation and reduction of artifacts can also be seen regarding ablation results. Taken together, it is the first piece connecting the fields of model-based regularization and modern deep learning, thereby providing a clinical pipeline toward interpretable, high-fidelity, and deployable medical image reconstruction.

Contravariant reich type F-contraction in F-bipolar metric spaces with application

2025-04-11T11:22:41+03:00

In this work, we explore fixed point (FP) results using the concept of contravariant Reich-type F -contraction within the framework of F -bipolar metric spaces (F-BMS). Our findings extend certain results from the existing literature. Additionally, we provide an example and an application to demonstrate the existence and uniqueness of the integral equation (IE).