The primary goal of this work is to introduce dynamic graphs. Specifically, it will demonstrate that the matrix is the basic matrix of interconnections (adjacency). It functions to explain a particular graph D's nominal structure based on the presumption that graph D's lines' functional dependence that is, the matrix E's elements (edges) are arranged so, that equation. It can be obtained by two scalar equations and described the evolution of the dynamic matrix E over time. To transform nonlinear differential equations derived from delay differential equations (DDEs) to linear differential equations, the purpose of using a dynamical graph. With this method, we applied on the biological problem of Lotaka-Volterra delay to studying stability by the backstepping method to delay differential equation (DDE) system to investigate stability on the impact of unsure interconnections between subsystems and solve it.

This study examines the impact of the non-Darcy Forchheimer coefficient, Activation energy, chemical reaction, thermal radiation, and Brownian parameter on the flow of a viscous Newtonian fluid undergoing Brownian motion across a stretching sheet with a changing heat source or sink. The model equation accounts for ohmic heating effects as well as changes in fluid viscosity and thermal conductivity. By using pertinent dimensionless variables, the flow equations are transformed. Also, the transformed nonlinear equations are solved by the MATLAB-BVP4C method. Graphs and tables are used to examine the influence of the relevant parameters. Our key findings are that the Forchheimer parameter and variable viscosity parameter decline fluid velocity. Brownian parameter and thermophoresis improve fluid temperature and chemical reaction parameter declines fluid concentration.

A novel numerical scheme is developed in this work to approximate solutions (APPSs) for nonlinear fractional differential equations (FDEs) governed by Robin boundary conditions (RBCs). The methodology is founded on a spectral collocation method (SCM) that uses a set of basis functions derived from generalized shifted Jacobi (GSJ) polynomials. These basis functions are uniquely formulated to satisfy the homogeneous form of RBCs (HRBCs). Key to this approach is the establishment of operational matrices (OMs) for ordinary derivatives (Ods) and fractional derivatives (Fds) of the constructed polynomials. The application of this framework effectively reduces the given FDE and its RBC to a system of nonlinear algebraic equations that are solvable by standard numerical routines. We provide theoretical assurances of the algorithm’s efficacy by establishing its convergence and conducting an error analysis. Finally, the efficacy of the proposed algorithm is demonstrated through three problems, with our APPSs compared against exact solutions (ExaSs) and existing results by other methods. The results confirm the high accuracy and efficiency of the scheme.

Rotating channel flows are an important part of scientific and engineering applications including microfluidic devices, rotating machinery, and energy-efficient thermal systems. Despite extensive research, the behavior of magnetized fluids, thermal-diffusive processes and rotational effects on non-Newtonian fluids has remained underexplored. This study investigates the flow of tangent hyperbolic fluid in a vertical rotating channel under the effects of magnetic field (Lorentz force), Coriolis force, Dufour and Soret effects. The governing nonlinear differential equations are solved using MATLAB’s bvp5c solver. Results show that increasing the Grashof number from 0.5 to 1.25 enhances primary velocity by 42% which is highly relevant in designing rotating heat exchangers where natural convection aids cooling efficiency, whereas a higher Hartmann number suppresses it by 22.10% due to magnetic damping, highlights how magnetic control can be employed in MHD pumps to regulate flow rates. A 4.46% decrease in temperature was observed with an enhanced Soret effect, indicating improved thermal regulation particularly beneficial in applications like thermal insulation systems and microelectronic cooling. The entropy generation increases by 2.39% with higher values of the pressure gradient parameter, indicating greater energy loss in pressure-driven microchannel flows often encountered in cooling systems and biomedical devices. The Bejan number, skin-friction coefficient, Nusselt number and Sherwood number behavior are also analyzed. The findings provide quantitative insights relevant to designing advanced cooling systems and chemical reactors.

This study examines the effect of chemical reaction and heat source/sink on steady two-dimensional mixed convective boundary layer flow of a hybrid nanofluid (HNF) over an inclined permeable plate/cylinder. The HNF is constructed by dispersing copper oxide (CuO) and titanium dioxide (TiO2) nanoparticles in water (H2O) as the base fluid. The model considers convective boundary conditions in both temperature and nanoparticle concentration. The resulting governing partial differential equations (PDEs) are reduced to a scheme of nonlinear ordinary differential equations (ODEs) via similarity transformations and numerically resolved by means of MATLAB’s bvp4c solver. Originality of this paper deceptions in integrating a numerical solver with an optimized feed-forward artificial neural network (FF-ANN) based on the Levenberg–Marquardt algorithm (LMA) to model HNF flow along with heterogeneous and homogeneous chemical reactions, heat source/sink, and inclination effects, a combination rarely explored in previous studies. The results indicate that porosity and inclination parameters reduce the velocity profiles, while increased concentration of nanoparticles and heat source/sink effect enhance thermal distribution. The LMA-ANN model possesses good predictive ability with the mean squared error (MSE) varying between 10−08 and 10−10. There is excessive consistency among the numerical solutions, as presented. The outcomes showcase the huge potential of HNFs and ANN-enhanced modeling to boost heat and mass transfer in complex engineering and industrial operations.

In this paper, we introduce a new LM parameter and incorporate a nonmonotone trust region technique to the modified Levenberg-Marquardt (MLM) method proposed by Fan [Mathematics of Computation, 81, 447–466, 2012]. We prove the global convergence and the local convergence rate of the new MLM method under the H o ¨ derian continuity of the Jacobian and the H o ¨ derian local error bound condition which are more general than the Lipschitz continuity of the Jacobian and the local error bound condition used by Fan. Moreover, we evaluate the numerical performance of the new MLM method by solving the discretized two-point boundary value problem, and some nonlinear equations arising in the field of complementary problems. Our numerical experiments provide preliminary indications of the local fast convergence rate and suggest potential improvements in computational efficiency compared to Fan's method.

The article explores the properties of gravitational vacuum stars, or gravastars, within the framework of $f(R,G,T)$ gravity, where $R$ represents the Ricci scalar, $G$ denotes the Gauss-Bonnet invariant, and $T$ stands for the trace of the energy-momentum tensor. Incorporating the effects of electric charge, the study employs suitable numerical techniques to solve the highly nonlinear differential equations governing the system. For a well-established cosmological model in $f(R,G,T)$ gravity, the internal structure of gravastars is analyzed in terms of three distinct regions: the core, the shell, and the exterior. The mass-radius relationship of the charged gravastar is graphically illustrated, highlighting its evolution within this modified gravity scenario.

Reservoir computing (RC) is promising for achieving low power consumption neuromorphic devices. In this study, we developed an all-solid-state electric double layer transistor using multilayer MoS2 to realize high-performance physical RC. We have demonstrated the high performance of a MoS2-based Raman-ion-gating reservoir, in which gate voltage-dependent resonant Raman scattering spectra of MoS2 were used as computational resources in addition to drain and gate current responses. Our device achieved good performance, such as >97% accuracy in various nonlinear waveform transformations and 97.8% accuracy in solving a second-order nonlinear dynamic equation. Information processing capacity was evaluated to elucidate the origin of the high performance of our system.

ABSTRACT The objective of the present study is to develop a PyTorch‐based deep neural network framework to predict velocity and temperature profiles in the calendering process of incompressible, non‐Newtonian fluids with temperature‐dependent viscosity. The governing partial differential equations are non‐dimensionalized using appropriate variables and simplified using the lubrication approximation theory, which reduces them to a system of nonlinear ordinary differential equations. Analytical solutions for pressure, velocity, and temperature fields are obtained using a perturbation method. Key engineering quantities, including detachment point, sheet thickness, roll separation force, power input, Nusselt number, and streamlines, are evaluated using the Regula Falsi method and numerical integration. Symbolic solutions are visualized to analyze the influence of various physical parameters. The artificial neural network model is developed in Python using PyTorch, employing sigmoid activation functions. Model performance is assessed through loss curves, absolute error analysis, and comparative bar plots. The framework achieves remarkable precision, with mean squared error values of for velocity profiles and for temperature profiles, with coefficients of determination for both cases. Furthermore, the influence of key parameters on convective heat transfer is analyzed through the Nusselt number. Results indicate that increasing the Weissenberg number enhances heat transfer, while a higher material parameter leads to its reduction. Additionally, an increase in the Brinkman number decreases both the sheet thickness and power input. This framework enables real‐time optimization of polymer sheet thickness, reduces roll separation forces in rubber processing, and facilitates energy‐efficient nanomaterial coating applications.

The time-elapsed model for neural assemblies is a nonlinear age-structured equation where the renewal term describes the network activity and influences the discharge rate, possibly with a delay due to the length of connections. We first solve a long standing question, namely that an inhibitory network without delay can promote desynchronization and stabilizes network activity by proving rigorously that the solution converges to a unique steady state. Our approach is based on the observation that a non-expansion property holds. However a non-degeneracy condition is needed and, besides the standard one, we introduce a new condition based on strict nonlinearity. When a delay is included, following previous works for Fokker-Planck models, we prove that the network can generate periodic solutions, both in inhibitory and excitatory networks. To this end, we introduce a new formalism to establish rigorously this property for large delays. Moreover, the fundamental contraction property can extend to other age-structured equations and systems.

Solving models comprised of nonlinear differential equations (DEs) in NONMEM using ADVAN6 or ADVAN13 typically requires substantially longer run times than models comprised of linear DEs, which in some cases allow for analytical solutions. Often the need to use nonlinear DE solvers results from pharmacokinetic (PK) variations over the dosing interval introducing the nonlinearity via a nonlinear transfer function, as is the case for indirect-response models and enzyme induction models. As long run times hinder model development, it is desirable to derive suitable approximations to speed up model solutions. The zero-order hold, a concept used in the field of advanced process control to optimize control decisions, provides an attractive approximation for these situations that often results in a sequential system of simpler DEs that in some cases can be solved analytically. Two examples, an indirect-response model and an enzyme induction model, demonstrate that the zero-order hold approximation provides a substantial reduction in computational time (up to ~ 140-fold) without unduly biasing the parameter estimates. These examples demonstrate that the zero-order hold approximation offers an attractive method for efficiently solving models where time-varying PK leads to a nonlinear system of DEs.

Positivity of global solution for a singular nonlinear heat equation associated with the Bessel operator

The paper is devoted to the study of the two-phase free boundary problem for nonlinear partial differential equations describing the evolution of a foam drainage in the one dimensional case which was proposed by Goldfarb et al. in 1988 in order to investigate the flow of a liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. In a series of papers, the authors have already solved the same problems without free boundary and with free boundary situated at the lower and the upper parts in the foam column, respectively. In this paper it is shown that the free boundary problem for the foam drainage equations with a sharp interface between dry and wet foams admits a unique global-in-time classical solution; this is done by a standard classical mathematical method, the maximum principle, and the comparison theorem. Moreover, the existence of the steady solution and its stability are shown.

Existence of Mild Solutions and Approximate Controllability of Stochastic Mixed Voltera - Fredholm Type Nonlinear Third-order Dispersion Equation

This paper explores the use of Riccati subequation neural networks to solve nonlinear partial differential equations modeling complex biological processes like cancer, brain function, and wound healing. These equations involve spatially varying biochemical signaling and tissue regeneration. The study applies a method where solutions to the Riccati problem are integrated into neural networks, with the first hidden layer neurons specifically providing the solution to the Riccati equation. Exact solutions for differential equations may be obtained using the suggested approach. In order to verify the mathematical foundation of this technique, we examine the proposed equations, which leads to the derivation of hyperbolic function solutions, trigonometric function solutions, and rational solutions. Using various graphical illustrations, the dynamic properties of certain solutions associated with waves have been demonstrated. The results provided in this study have the potential to improve understanding of the nonlinear dynamic characteristics displayed by the specified system and to confirm the effectiveness of the techniques that have been implemented.

In this study, we examined the steady movement of Carreau-Yasuda hybrid nanofluid across an irregularly extended sheet influenced by the magnetic field, non-Fourier heat flux, couple stress effects, and the synergistic impacts of Soret and Dufour phenomena. The governing partial differential equations are transformed into a system of nonlinear ordinary differential equations using appropriate similarity variables and then solved using the bvp4c solver. It is detected that the Nusselt number drops by 13.30% when the Dufour number falls between 0 to 1 and the rate of mass transmission drops by 9.69% when the Soret number falls between 0 to 1.

ABSTRACT In this study, we delve into the intricate dynamics of the () dimensional ‐deformed tanh‐Gordon equation, exploring its fractional form through the lens of the Caputo fractional derivative. By employing the innovative controlled Picard technique combined with Laplace transforms, we derive highly accurate approximate solutions, ensuring both convergence and reliability in the realm of fractional nonlinear equations. Our rigorous analysis of the solution's existence and uniqueness provides a robust theoretical foundation, while two‐ and three‐dimensional graphical representations vividly illustrate the profound impact of fractional and ‐deformed parameters on the system's behavior. The findings not only highlight the efficacy of the controlled Picard technique in tackling complex fractional models but also offer valuable insights into their numerical treatment. This research opens new avenues for exploring higher dimensional and coupled fractional systems, paving the way for future advancements in nonlinear dynamics and fractional calculus.

Abstract In this study, we investigate a (3+1)-dimensional generalized nonlinear wave equation modeling bubbly liquid systems using the non-classical symmetry method. Distinct classes of symmetries are identified and systematically categorized, leading to a variety of novel and exact solutions. Beyond the analytical results, we conduct a comprehensive dynamical analysis, including bifurcation structures, equilibrium points, and chaotic behavior. Chaos diagnostics are carried out using phase portraits, Poincaré maps, power spectra, Lyapunov exponents, bifurcation diagrams, and return maps. The novelty of this study lies in deriving previously unreported families of exact solutions for the (3+1)-dimensional bubbly liquid wave equation through non-classical symmetries, and in coupling these analytical findings with a systematic bifurcation and chaos analysis. This combined framework provides new insights into the nonlinear dynamics of bubbly liquids that have not been addressed in earlier studies. The findings reveal new families of exact solutions and complex nonlinear behaviors, offering valuable insights into the transition mechanisms between ordered and chaotic phases in wave propagation within bubbly liquids.

In this study, we investigate the Lonngren wave equation by means of the generalized the -expansion method to obtain exact traveling-wave solutions for a variety of parameter regimes. This technique produces fresh analytical expressions that reveal the equation’s dynamic behavior. We verify the correctness of these solutions by substituting them back into the original equation. In addition, three-dimensional surface plots and contour diagrams are presented to illustrate the physical characteristics of the resulting waveforms. Overall, our results demonstrate the effectiveness of the generalized expansion approach for solving nonlinear partial differential equations and deepen the theoretical understanding of wave propagation in applied settings.

In the so-called Child–Langmuir law, established since 1911, an electron beam is formed linking two electrodes, which are assumed to be two parallel plates of area A, separated by a finite distance D. When D≪A, “edge effects” are negligible and the modelling is reduced to a nonlinear boundary problem for a singular ordinary differential equation in which a constant coefficient (the generated electric current j) must be found in order to get simultaneously Dirichlet and Neumann homogeneous boundary conditions in one of the extremes. If D>A, then the problem becomes much more difficult since the “edge effects” arise in the plane (x,y) and the electric current (now j(x) due to the presence of a very large perpendicular magnetic field) must be determined in order to get solutions u(x,y) of a singular semilinear equation which are partially flat (u=∂u ∂n=0 on a part of the boundary). In this paper, we offer a rigorous mathematical treatment of some former studies (Joel Lebowitz and Alexander Rokhlenko (2003) and Alexander Rokhlenko (2006)), where several open questions were left open: for instance, the need for a singularity of j(x) near the cathode edge to get such partially flat solutions.