Numerical solutions for second‐order parabolic partial differential equations (PDEs), specifically the nonlinear heat equation, are investigated with a focus on analyzing residual corrections. Initially, the Galerkin weighted residual method is employed to rigorously formulate the heat equation and derive numerical solutions using third‐degree Bernstein polynomials as basis functions. Subsequently, a proposed residual correction scheme is applied, utilizing the finite difference method to solve the error equations while adhering to the associated error boundary and initial conditions. Enhanced approximations are achieved by incorporating the computed error values derived from the error equations into the original weighted residual results. The stability and convergence of the residual correction scheme are also analyzed. Numerical results and absolute errors are compared against exact solutions and published literature for various time and space step sizes, demonstrating the effectiveness and precision of the proposed scheme in achieving high accuracy.

This study uses the Xue model to explore how well a nanofluid transfers heat in a steady oblique stagnation‐point flow. It examines the impact of nonlinear thermal radiation on a mixture of three different nanoparticles as the fluid moves along a stretching surface. This intended comparison model is unique and still scarce in the literature. Trihybrid nanofluids or composites have, therefore, been created to enhance heat transfer efficiency. Three different types of nanoparticles (Fe3O4, Cu, and TiO2) are exploring circumstances where ethylene glycol is the base medium. A mathematical framework is developed. Using the appropriate transformations, the system of partial differential equations (PDEs) is transformed into an ordinary differential system of three equations (ODEs), which is evaluated numerically using the bvp4c method. This integrated technique facilitates the convergence process effectively. A detailed analysis is conducted of the graphical representation and the physical behavior of important factors. On temperature and velocity profiles, the impacts of several variables, including a thermal radiation, surface heating parameter, stretching ratio, and particle volume fraction, are investigated thoroughly. The results show that the (Fe3O4 + Cu + TiO2)/ethylene glycol nanofluid outperforms with a high particle volume fraction of TiO2. It has been demonstrated that (Fe3O4 + Cu + TiO2)/ethylene glycol nanofluid with a high particle volume fraction of TiO2 has considerably greater thermal radiation than other nanoparticles.The inclusion of nanofluids significantly improves heat transfer compared with conventional fluids due to their higher thermal conductivity, which is crucial for enhancing heat dissipation at stagnation points in solar systems.

This paper analyzes a modified Leslie–Gower predator–prey model incorporating fear and refuge effects on the prey, along with hunting cooperation among predators and linear harvesting of the predator population. The model’s dynamical properties including boundedness, positivity, extinction criteria, and the existence and stability (both local and global) of equilibrium points are rigorously examined. Bifurcation analysis is also conducted to explore Hopf, transcritical, and saddle‐node bifurcations. The study demonstrates that fear has a stabilizing effect on the system dynamics, while prey refuge, hunting cooperation, and harvesting can induce instability. Analytical findings are supported by extensive numerical simulations, which illustrate how variations in these ecological factors influence population dynamics and bifurcation thresholds. The findings provide insight into the intricate interactions between ecological and behavioral elements in predator–prey relationships.

In order to solve fractional differential equations on quantum domains, this work provides a spectral approach based on higher‐order ( q , τ )‐Bernoulli functions and polynomials. We build a robust basis for approximation in ( q , τ )‐weighted Hilbert spaces by using the orthogonality properties of these extended polynomials and the Sheffer‐type generating function. Prototype equations of the form D q , τ u ( x ) = f ( x ) are numerically solved using the ( q , τ )‐Lagrange interpolation approach modified to represent arbitrary functions in terms of Bernoulli bases. Spectral expansion is used to recreate the solution, and a thorough example is given. The technique shows spectral convergence and shows how well higher‐order ( q , τ )‐Bernoulli systems capture the global structure and local behavior of fractional quantum calculus solutions.

In this paper, an algebraic decay rate for the radius of spatial analyticity of solutions to the Kawahara equation is investigated. With given analytic initial data having a fixed radius of analyticity σ0, we derive an algebraic decay rate σ(t) ~ |t|−1/2 for the uniform radius of spatial analyticity of solutions to the Kawahara equation. This improves a recent result due to Ahn et al.’s study, where they demonstrated a decay rate of order |t|−1. Our strategy mainly relies on an approximate conservation law in a modified Gevrey space and bilinear estimate in Bourgain space.

The purpose of this article is to present some fixed point theorems to guarantee the existence and uniqueness of common fixed points for two mappings (not necessary continuous), satisfying generalized contractions involving rational expressions in the setting of extended parametric Sb‐metric spaces. To substantiate our findings, some examples with graphical representation are also given. Moreover, as an application of our findings, the existence and uniqueness of common solution to the system of integral equations followed by an example are presented.

This paper investigates the Fuzzy Adomian Decomposition Method to find approximate analytical solutions for linear and nonlinear fuzzy Darboux problems using the Caputo‐type mixed fractional derivative, which plays an important role in applied and engineering sciences. The solutions are formulated as series with easily calculable terms. Multiple examples are included to illustrate the effectiveness of the approach, which employs the σ ‐level representation of fuzzy numbers. The results are presented graphically, depicting both the lower and upper bounds of the solutions.

This paper investigates linearization methods used in the development of an adaptive block hybrid method for solving first‐order initial value problems. The study focuses on Picard, linear partition, simple iteration, and quasi‐linearization methods, emphasizing their role in enhancing the performance of the adaptive block hybrid method. The efficiency and accuracy of these techniques are evaluated through solving nonlinear differential equations. The study provides a comparative analysis focusing on convergence properties, computational cost, and the ease of implementation. Nonlinear differential equations are solved using the adaptive block hybrid method, and for each linearization method, we determine the absolute error, maximum absolute error, and number of iterations per block for different initial step‐sizes and tolerance values. The findings indicate that the four techniques demonstrated absolute errors, ranging from O(10−12) to O(10−20). We noted that both the Picard and quasi‐linearization methods consistently achieve the highest accuracy in minimizing absolute errors and enhancing computational efficiency. Additionally, the quasi‐linearization method required the fewest number of iterations per block to achieve its accuracy. Furthermore, the simple iteration method required fewer number of iterations than the linear partition method. Comparing minimal CPU time, the Picard method required the least. These results address a critical gap in optimizing linearization techniques for the adaptive block hybrid method, offering valuable insights that enhance the precision and efficiency of methods for solving nonlinear differential equations.

This study introduces a new three‐dimensional chaotic oscillator system characterized by zero eigenvalues, with stability localized in the center manifold, an uncommon feature in chaotic system design. The proposed system is constructed entirely from nonlinear terms and demonstrates complex dynamics validated through bifurcation analysis and Lyapunov exponent computation. The results of this work are the application of the system to model the dynamics of the shadow economy, where the variables represent corruption, enforcement, and hidden economic activity. The model captures the unpredictable feedback interactions inherent in such systems and illustrates how minor changes in parameters lead to vastly different long‐term outcomes. Furthermore, a fractional‐order extension of the system is investigated using the Caputo derivative to determine the effects of memory in chaotic evolution. Numerical simulations reveal that fractional orders significantly influence attractor behavior, with transitions from chaos to regular dynamics. This paper contributes a structurally novel chaotic model, a fractional‐order analysis framework, and an application to economic dynamics—providing valuable insights into both chaos theory and economic system modeling.

This paper explores the solvability of multiterm hybrid functional equations with multiple delays, addressing these equations under some nonlocal hybrid boundary conditions. By applying Schauder fixed‐point theorem, we establish the existence of continuous solutions and provide sufficient requirements for the continuous dependence of the unique solution on some factors. In addition, the existence of integrable solutions is examined, broadening the theoretical applicability of these results. Finally, to demonstrate the utility of the approach, an example is provided, showcasing the effectiveness of the proposed method in handling several hybrid problems.