ANALYTICAL AND NUMERICAL SOLUTIONS FOR NONLINEAR EQUATIONS
Year:2024 | Volume:9 | Issue:2|Papers Count:12

In this research, we consider the linear and nonlinear Volterra integral equations (VIEs). The main aims of research is to approximate the integral by Gauss-Tur$\acute{a}$n quadrature rule and then using extended cubic B-spline as the bases function. The unknown coefficients in combination determine by collocation method. The arising system of linear and nonlinear can be solved via iterative method. Error analysis is investigated theoretically. Numerical text problems are considered to justify the applicability and efficient nature of our approach, comparison of the results justify the considerable accuracy and efficiency proposed methods. The extended parameter in valued in the spline can be chosen in such a way to  improve the accuracy also.

The sparsity of the Gram matrix in linear regression can influence the model's accuracy. Sparse matrices reduce computational complexity and improve generalization by minimizing overfitting. This advantage is particularly beneficial in high-dimensional data where the number of features exceeds the number of observations. This paper explores the integration of Radial Basis Functions (RBFs) in developing sparse Gram matrix fuzzy regression models. RBFs are powerful tools for function approximation, defined by their dependence on the distance from a center point, which allows for flexible modeling of nonlinear relationships. The focus will be on compactly supported RBF kernels, which facilitate sparsity in the Gram matrix, thereby improving computational efficiency and memory usage. By leveraging the properties of RBFs, particularly their ability to localize influence and reduce dimensionality, we aim to enhance the performance of fuzzy regression models. This study will present theoretical insights and empirical results demonstrating how the adoption of RBFs can lead to significant improvements in model accuracy and computational speed, making them a valuable asset in the field of fuzzy regression analysis.

In this paper, we investigate the real roots of a special class of square matrices, leveraging the properties of involutory and idempotent matrices. We focus on determining real roots for real orthogonal and symmetric matrices, demonstrating how involutory matrices facilitate this process. Our results show that a real involutory matrix of order $n$ with a positive determinant always admits a real root. Furthermore, for real symmetric matrices, we establish that a real root exists if every negative eigenvalue appears with even multiplicity. We also explore the structure of idempotent matrices, presenting a general block form derived through similarity transformations. Specifically, we prove that for invertible submatrices $A$ and $D$, along with arbitrary block matrices $B$ and $C$, a constructed matrix $P$ exhibits idempotency. An illustrative example is provided to clarify this construction, highlighting its application in generating idempotent and involutory matrices from simpler components. Additionally, we examine the root-approximability of orthogonal matrices, showing that certain sequences of matrices converge to the identity while their powers approximate the original matrix. This work extends existing results on matrix functions and diagonalization, offering practical insights into the analysis and computation of matrix roots. Our findings contribute to the broader understanding of matrix theory, with potential applications in numerical linear algebra and functional analysis.

In this paper, ‎the Homotopy Perturbation Method (HPM) is employed to obtain approximate semi-analytical solutions for Fuzzy Initial Value Problems (FIVPs) within the framework of generalized differentiable. ‎The original FIVP is reformulated as a pair of parameterized ordinary differential equations, ‎which are then solved iteratively using HPM‎. ‎Numerical results show that the approximate solutions converge rapidly to the exact fuzzy solutions, ‎achieving high accuracy even with a limited number of perturbation terms. ‎These findings underscore HPMs effectiveness as a simple yet powerful technique for addressing fuzzy differential equations. ‎Moreover, ‎the methods flexibility indicates its potential for solving higher-order and more complex fuzzy differential systems. ‎Recent studies including the application of HPM to fuzzy impulsive fractional differential equations under generalized Hukuhara differentiable, ‎as well as hybrid and transform-based extensions such as the Elzaki Transform Homotopy Perturbation Method (ETHPM) further highlight the evolving scope and versatility of HPM in fuzzy problem-solving contexts.

The Fokker–Planck equation is widely used to describe how systems evolve when randomness plays a role. It appears in many fields, including physics, finance, biology, and engineering. Classical numerical methods usually require discretization, which can make the computation expensive, unstable, or less accurate. In this work, we present a direct method for solving these equations using Least Squares Support Vector Regression (LSSVR) with Legendre kernels. Our approach avoids discretization and provides global optimization, which helps overcome the difficulties faced by loss-based methods such as Physics-Informed Neural Networks (PINNs). The use of Legendre kernels gives strong approximation properties and ensures high accuracy in the solutions. We tested the method on several problems and found that it achieves very precise results while being faster and more stable than PINNs. To further improve reliability, we also applied automatic hyperparameter tuning, which adapts the method to each problem without manual adjustment. These results show that LSSVR with Legendre kernels is a simple, accurate, and efficient tool for scientists and engineers who need to solve Fokker–Planck equations.

This paper presents a novel and efficient fully discrete numerical scheme for distributed-order fractional partial differential equations involving both the Caputo time-fractional derivative and the Riesz space-fractional derivative. Such equations frequently arise in the modeling of anomalous diffusion and transport phenomena, where accurate and stable computational methods are crucial. The temporal discretization is carried out using the second-order generalized L1 (gL1-2) scheme, which improves accuracy over traditional L1-based methods. For the spatial discretization, the Riesz derivative is approximated by a second-order finite-difference method, ensuring robustness and precision. The resulting scheme provides a high-order numerical framework that can effectively address a wide class of distributed-order fractional models. A rigorous theoretical analysis is conducted, proving unconditional stability and optimal convergence rates via the energy method. The effectiveness of the scheme is further validated through two numerical experiments, which confirm the theoretical results and highlight the computational efficiency, accuracy, and practical applicability of the proposed approach.

The increase in urban population, change in lifestyle, change in diet, as well as increase in the level of well-being and living standards in urban communities have caused a large amount of solid waste in big cities. Currently, managing the process of solid waste in big cities is one of the most important problems in developing countries. Most of the studies in the literature are focused on reverse logistics for one type of product for the recovery or recycling process, and not much attention has been paid to the reuse distribution network through charities. In this research, a framework for reusing all kinds of household appliances to reduce urban solid waste and help low-income families is proposed. A mixed integer linear mathematical model with uncertainty in the number of products is presented for reverse logistics network optimization. This model has been solved by used Genetic Algorithm and its applications have been discussed. In the designed logistics network, various topics such as recycling, repair and charity centers are considered. In order to show the performance of the presented model, a numerical example has been solved by software MATLAB. In this example, the software MATLAB obtains the network structure at an optimal cost. The results confirm the applicability of the model by providing a large number of second-hand products that can be transported and reused at an affordable cost.

This work is devoted to a rigorous analysis of the existence and uniqueness of solutions for a class of high-order nonlinear differential equations of fractional order. The considered problem is defined by a Caputo fractional derivative and is augmented by a set of nonlocal boundary constraints. A key feature of these constraints is an integral condition that couples the behavior of the solution across its entire spatial domain, reflecting a global dependency. Our primary analytical strategy is to recast the differential problem as a fixed-point equation for an equivalent integral operator. This is accomplished by first methodically constructing the Green's function associated with the corresponding linear problem. With the integral operator established, the existence of a unique solution for the full nonlinear problem is then proven by leveraging the power of the Banach contraction mapping principle. To demonstrate the practical relevance and applicability of our theoretical framework, a detailed illustrative example is presented and analyzed.

We investigate a nonlinear equation of state arising in a phenomenological model of dark energy. The model is motivated by modifications to quintessence dynamics and leads to a non-linear differential equation for the Hubble parameter. We provide both approximate analytical methods and numerical solutions, demonstrating how nonlinearity alters the late-time acceleration of the Universe. The results suggest that deviations from linearized treatments can significantly modify the effective equation of state parameter at redshifts $z < 1$, with potential implications for precision cosmology.

The study of black hole thermodynamics has revealed profound connections between gravitation, quantum theory, and statistical mechanics. In many instances, the key physical information is encoded in nonlinear algebraic or transcendental equations that relate horizon radius, temperature, and pressure. In this work, we examine a specific nonlinear equation arising from the extended phase space of charged anti-de Sitter (AdS) black holes. By analyzing its structure and obtaining approximate and exact solutions, we highlight the physical implications for the thermodynamic stability of black holes. Our results clarify the role of nonlinearities in determining critical points and phase transitions analogous to the van der Waals fluid.

We investigate a class of non-linear differential equations arising from five-dimensional minimal gauged supergravity. Specifically, we focus on the structure of the scalar function governing supersymmetric backgrounds and show how its governing equation can be reduced to a solvable non-linear system. By employing an analytic approximation and numerical methods, we present explicit solutions and analyze their physical interpretation within the AdS/CFT framework. Our results highlight the interplay between geometry, gauge fields, and supersymmetry, and extend the known catalogue of exact solutions.

We investigate nonlinear differential equations arising in holographic models of black hole spacetimes within the AdS/CFT correspondence. Emphasis is placed on analytical approximations and numerical solutions of scalar perturbations and nonlinear wave equations in asymptotically anti-de Sitter black holes. We show how effective potentials lead to boundary value problems governed by nonlinear ordinary differential equations. Analytical methods, such as matched asymptotic expansions and perturbative ans"{a}tze, are compared with spectral numerical approaches. Applications to holographic thermalization, quasinormal spectra, and cosmological analogues are discussed. Our results highlight the interplay between nonlinear analysis and physical interpretation in string-inspired models of gravity.