Differential Equations Research Articles

Impact of Thermal Energy on Water and Ethylene Glycol (50:50)-Based Rotating Hybrid Nanofluid Past A Riga Surface with Radiation, Homogeneous and Heterogeneous Reactions

Hybrid nanofluid is crucial in improving the efficiency of heat transmission in thermal engineering applications, particularly the cooling of electrical equipment, heat exchangers, fuel cells, printing machines, etc. This study sought to investigate the augmentation of heat and mass transmission by the use of a rotating hybrid nanofluid consisting of a mixture of gold and silver nanoparticles suspended in water and ethylene glycol (50:50) past a heated Riga surface with velocity slip and suction. The energy equation is constructed using nonlinear radiation and heat consumption. The impact of both homogeneous and heterogeneous processes on the hybrid nanofluid is also explored. The governing mathematical model begins with partial differential equations that are converted into ordinary differential equations using a suitable conversion technique. The results of numerical computations are recorded as graphs and tables using the bvp4c scheme in MATLAB. It is noticed that both directional velocities undergo significant negative changes as a result of enhanced values of the rotational parameter. The higher levels of the radiation parameter resulted in significant enhancements in the thermal profile. The falling behavior of the nanoparticle concentration profile is recorded against a greater quantity of both chemical reaction parameters. Based on our analysis, when the Biot number changes from 0.4 to 0.8, the most significant heat transmission gradient occurs.

Numerical solutions and stability analysis of hybrid Casson nanofluid flow with MHD and heat transfer effects

AbstractThis research addresses the complex dynamics of hybrid Casson nanoliquids in flow and heat transfer applications, focusing on the interaction between fluid dynamics and thermal phenomena in the presence of magnetohydrodynamics (MHD), viscous dissipation, and Joule heating. The motivation behind this study stems from the need to enhance the efficiency of heat transfer processes in various engineering applications, such as cooling systems and electronic devices, where hybrid nanofluids can offer superior thermal performance compared to conventional fluids. The novelty of this work lies in its comprehensive numerical investigation of a hybrid Casson nanoliquid over a moving permeable surface, incorporating a detailed analysis of MHD effects, viscous dissipation, and Joule heating. Using the Tiwari and Das model to formulate the governing equations, the study employs similarity transformations to convert these equations into a system of ordinary differential equations (ODEs). MATLAB is then used to derive numerical solutions, with a stability analysis ensuring the physical dependability of these solutions. Key findings reveal that the temperature distribution of the hybrid nanoliquid shows a positive correlation with both Prandtl and Hartmann numbers. Additionally, a positive relationship between temperature and the Eckert number is observed. These insights offer valuable guidance for engineers aiming to optimize heat transfer processes using hybrid nanofluids, highlighting their potential for improved thermal management in practical applications.

Approximation Properties of Chlodovsky-Type Two-Dimensional Bernstein Operators Based on (p, q)-Integers

In the present study, we introduce the two-dimensional Chlodovsky-type Bernstein operators based on the (p,q)-integer. By leveraging the inherent symmetry properties of (p,q)-integers, we examine the approximation properties of our new operator with the help of a Korovkin-type theorem. Further, we present the local approximation properties and establish the rates of convergence utilizing the modulus of continuity and the Lipschitz-type maximal function. Additionally, a Voronovskaja-type theorem is provided for these operators. We also investigate the weighted approximation properties and estimate the rate of convergence in the same space. Finally, illustrative graphics generated with Maple demonstrate the convergence rate of these operators to certain functions. The optimization of approximation speeds by these symmetric operators during system control provides significant improvements in stability and performance. Consequently, the control and modeling of dynamic systems become more efficient and effective through these symmetry-oriented innovative methods. These advancements in the fields of modeling fractional differential equations and control theory offer substantial benefits to both modeling and optimization processes, expanding the range of applications within these areas.

SEGODE: a structure-enhanced graph neural ordinary differential equation network model for temporal link prediction

SEGODE: a structure-enhanced graph neural ordinary differential equation network model for temporal link prediction

Some Exact Green Function Solutions for Non-Linear Classical Field Theories

We consider some non-linear non-homogeneous partial differential equations (PDEs) and derive their exact Green function solution as a functional Taylor expansion in powers of the source. The kind of PDEs we consider are dispersive ones where the exact solution of the corresponding homogeneous equations can have some known shape. The technique has a formal similarity with the Dyson–Schwinger set of equations to solve quantum field theories. However, there are no physical constraints. Indeed, we show that a complete coincidence with the statistical field model of a quartic scalar theory can be achieved in the Gaussian expansion of the cumulants of the partition function.

The influence of lymphatic vessels on nanoparticle distribution and heat transfer within tissue

AbstractThis study analytically investigates the dynamics of nanoparticle transport within a three‐dimensional porous cylinder simulating a lymphatic vessel, without external heat sources. The governing equations and boundary conditions are transformed to yield a system of ordinary differential equations, which are solved numerically using MATLAB built‐in function, bvp4c. Key parameters are visually examined and physically interpreted in relation to temperature, velocity, concentration, and Nusselt number profiles. The study reveals that the distribution of temperature and Nusselt number are maximized by increasing the heat transfer coefficient, whereas NP concentration is increased by decreasing it. Furthermore, the Brownian motion parameter enhances both heat transmission and NP concentration. It is also observed that simpler extravasation into lymphatics decreases tissue nanoparticle levels and heat conduction. Ultimately, optimal intra‐lymphatic nanoparticle distribution pathways are achieved by specifically varying heat transfer and interstitial mass flux patterns. By simulating biological barriers and lymphatic drainage, this model enhances our understanding of the underlying transport mechanisms controlling nanoparticle mobilization.

Construction of optimised theoretical model using ANOVA -Taguchi methodology for transient flow of Carreau nanofluid through microchannel prone to radiation

The current study intends to predict the optimised condition to attain the objective of acquiring highest heat transfer rate to develop an efficient model. The transient flow of Carreau nanofluid within a microchannel when channel walls are susceptible to radiation is contemplated. Buongiorno model is employed, which emphasizes the repercussions of Brownian motion and thermophoresis phenomena; also, mixed-convective flow is accounted. The modelled problem gives rise to partial differential equations, which are non-dimensionalized employing non-dimensional quantities. The resultant equations are solved numerically using the finite difference method. Results of analysis demonstrate that the Weissenberg number for n<1 depicts shear thinning nature, and for n>1, depicts shear thickening nature, decreasing velocity. The skin friction coefficient increases when solutal Grashof number rises for the high range of the Reynolds number. The Sherwood number increases when Schmidt number is less for increased value of Reynolds number. Optimization method reveals the highest heat transfer rate of 7.3687 for the considered model. ANOVA results show that the manipulation of Reynolds number is crucial with 57.29% impact and the manipulation of Prandtl number has minor impact of 1.41%on Nusselt number. Shear thinning nature of Carreau fluid finds its application in extrudability, printability and injectability and shear thickening nature is extensively used in industrial polishing, explosion resistance.

On entire solutions of certain partial differential equations

On entire solutions of certain partial differential equations

On a discrete version of Fejér inequality for α-convex sequences without symmetry condition

Abstract In this study, we introduce the notion of α \alpha -convex sequences which is a generalization of the convexity concept. For this class of sequences, we establish a discrete version of Fejér inequality without imposing any symmetry condition. In our proof, we use a new approach based on the choice of an appropriate sequence, which is the unique solution to a certain second-order difference equation. Moreover, we obtain a refinement of the standard (right) Fejér inequality for convex sequences.

Computational study with neural networks to double diffusion in Prandtl thermal nanofluid flow adjacent to a stretching surface design with numerical treatment

The study focuses on the implementation of neural networks in analyzing the behavior of Prandtl nanofluid near a extending surface in the existence of dual diffusion. The research investigates the merged consequence of thermal and concentration gradients on the fluid flow and heat transfer characteristics using advanced computational techniques. The investigation delves into the phenomenon of double diffusion in the flow of Prandtl nanofluid near a stretching surface (DD-PNSS), employing the Levenberg-Marquardt scheme with Artificial Neural Networks (LMS-ANNs). The application of similarity variables converts the non-linear partial differential equations into non-linear ordinary differential equations. Through the application of the Lobatto IIIa formula in a three-stage process, various data sets are generated for the LMS-ANNs by varying parameters such as the Prandtl fluid parameter ([Formula: see text]), Prandtl number (Pr), Brownian motion parameter (Nb), thermophoresis parameter (Nt), and Dufour-solutal Lewis number (Ld). The range of parameter α is from 2 to 3.5, parameter β from 0.1 to 0.9, Nb is from 0.1 to 0.4, Nt is from 0.1 to 0.7, Le is from 1 to 3 and Ld is from 0.1 to 0.5. The intrinsic behavior of embedded parameters on dimensionless velocity, temperature, solutal concentration, and nanoparticle concentration profiles is graphically evaluated. The proposed LMS-ANNs model is meticulously tested, validated, and trained using a multi-stage approach, and its performance is compared to established references to ensure its reliability. The effectiveness of the suggested LMS-ANNs model is further affirmed through regression analysis, Mean Squared Error (MSE) evaluation, and histogram studies, showcasing an exceptional accuracy level ranging from 10−08 to 10−10, setting it apart from alternative approaches and reference models.

The Performance of Fixed Step Size and Adaptive Step Size Numerical Methods for Solving Deterministic Cell-Growth Models

Deterministic cell-growth models describe the growth of cell populations using fixed mathematical rules, assuming no randomness in the system. These models are often based on differential equations that account for the rates of cell division, death, and other biological processes. The solution to the system is obtained via numerical methods. Most of the developed approaches are based on fixed step sizes. However, fixed step size implementation failed to offer the optimal solutions when dealing with stiff challenges. Fixed step size methods can be unstable for stiff equations, where some components of the solution change much more rapidly than others. The step size, required to maintain stability can become impractically small. Thus, the adaptive step size method is required. Adaptive step size methods adjust the step size dynamically based on the behavior of the solution, aiming to maintain a desired level of accuracy while optimizing computational efficiency. These methods are particularly useful for solving ordinary differential equations (ODEs) where the solution can vary rapidly in some regions and slowly in others. This study is devoted to comparing the implementation of fixed step size and adaptive step size in solving ordinary differential equations (ODEs). The fixed step size and adaptive step size numerical method are solved in this study via the fourth order Runge-Kutta method (RK4) and Runge-Kutta Fehlberg 45 method (RKF45). The performance of both numerical methods used will be analyzed by comparing the numerical results approximated with the actual data. Subsequently, the absolute error, relative error, and rounding-off error will be calculated to compare both approaches. Based on the more precise findings, this work has shown that adaptive step size is predicted to be the optimal representation for solving ODEs. As a result, this may help mathematicians to choose the most effective numerical approach for solving ODEs.

Mathematical Approach for Directly Solving Air–Water Interfaces in Water Emptying Processes

Emptying processes are operations frequently required in hydraulic installations by water utilities. These processes can result in drops to sub-atmospheric pressure pulses, which may lead to pipeline collapse depending on soil characteristics and the stiffness of a pipe class. One-dimensional mathematical models and 3D computational fluid dynamics (CFD) simulations have been employed to analyse the behaviour of the air–water interface during these events. The numerical resolution of these models is challenging, as 1D models necessitate solving a system of algebraic differential equations. At the same time, 3D CFD simulations can take months to complete depending on the characteristics of the pipeline. This presents a mathematical approach for directly solving air–water interactions in emptying processes involving entrapped air, providing a predictive tool for water utilities. The proposed mathematical approach enables water utilities to predict emptying operations in water pipelines without needing 2D/3D CFD simulations or the resolution of a differential algebraic equations system (1D model). A practical application is demonstrated in a case study of a 350 m long pipe with an internal diameter of 350 mm, investigating the influence of air pocket size, friction factor, polytropic coefficient, pipe diameter, resistance coefficient, and pipe slope. The mathematical approach is validated using an experimental facility that is 7.36 m long, comparing it with 1D mathematical models and 3D CFD simulations. The results confirm that the derived mathematical expression effectively predicts emptying operations in single water installations.

Generalized Laplace Transform with Adomian Decomposition Method for Solving Fractional Differential Equations Involving ψ-Caputo Derivative

In this study, we introduced the ψ-Laplace transform Adomian decomposition method, which is a combination of the efficient Adomian decomposition method with the generalization of the classical Laplace transform to treat fractional differential equations with respect to another function, ψ, in the Caputo sense. To validate the effectiveness of this method, we applied the derived recurrent scheme of the ψ-Laplace Adomian decomposition on several test numerical problems, including a real-life scenario in pharmacokinetics that models the movement of drug concentration in human blood. The solutions obtained closely matched the known solutions for the test problems. Additionally, in the pharmacokinetics case, the results were consistent with the available physical data. Consequently, this method simplifies the verification of numerous related aspects and proves advantageous in solving various ψ-fractional differential equations.

Quadratic constraint consistency in the projection-free approximation of harmonic maps and bending isometries

We devise a projection-free iterative scheme for the approximation of harmonic maps that provides a second-order accuracy of the constraint violation and is unconditionally energy stable. A corresponding error estimate is valid under a mild but necessary discrete regularity condition. The method is based on the application of a BDF2 scheme and the considered problem serves as a model for partial differential equations with holonomic constraint. The performance of the method is illustrated via the computation of stationary harmonic maps and bending isometries.

Study on heat transfer law of low temperature oxidation of coal under convection condition

Coal oxidation in the goaf generates heat accumulation influenced by airflow, leading to heat transfer within the porous coal structure. This study experimentally investigates the heat transfer characteristics of coal under convective conditions. A temperature migration rate measurement device was developed, and a formula for calculating the temperature migration rate was derived using the steady-state heat conduction differential equation and experimental data. The study examines the effects of temperature and airflow on the temperature migration rate and heat transfer characteristics. The experimental results indicate that the heat transfer effect of coal at low temperatures is minimal and volatile. As the temperature increases, efficiency improves, and heat transfer stabilizes. Airflow facilitates coal's heat transfer, causing the temperature generated by coal oxidation to concentrate on the downwind side. Additionally, airflow inhibits coal oxidation at low temperatures, while high temperatures promote it. Furthermore, the temperature migration rate of coal decreases with increasing temperature, initially decreases and then increases with rising airflow at low temperatures, whereas the opposite trend is observed at high temperatures.

An Efficient Analysis of Amplitude and Phase Dynamics in Networked MEMS-Colpitts Oscillators

Abstract This paper explores the interactions of both phase and amplitude in a network of N MEMS-Colpitts oscillators that are resistively coupled. The numerical simulations of the extensive networks of oscillators, required for emerging applications such as clock synchronization and neuromorphic computing, become computationally prohibitive as the number of oscillators increases. This complicates the design and evaluation of such systems, as understanding the effects of changes in coupling and design parameters can require many simulations. This study employs the method of multiple scales (MS) in combination with the harmonic balance method to convert the coupled differential equations governing the system of oscillators into a set of nonlinear evolution equations for the amplitude and phase of the oscillators. The amplitude and phase evolve on a timescale that is slow, commensurate with the damping in the system, compared with the fast timescale of the oscillation frequencies. The approach used in this study to describe the amplitude and phase dynamics offers significant computational efficiency (gains of 10× to 50× are shown) compared to direct numerical integration while maintaining an accurate representation of the response. The results of the presented simulations demonstrate the effect of coupling strength on the dynamics of the network, accounting for both phase and amplitude dynamics.

Opinion response functions are key to understanding the tipping of social conventions

Abstract The extent to which committed minorities can overturn social conventions is an active area of research in the mathematical modeling of opinion dynamics. Researchers generally use simulations of agent-based models (ABMs) to compute approximate values for the minimum committed minority size needed to overturn a social convention. In this manuscript, we expand on previous work by studying an ABM’s mean-field behavior using ordinary differential equation models and a new tool, namely opinion response functions (ORFs). Using this method allows for formal analysis of the deterministic model, which can provide a theoretical explanation for observed behaviors, e.g. coexistence or overturning of opinions. In particular, ORFs are a method for characterizing equilibria in our social model. Our analysis confirms earlier numerical results and supplements them with a precise formula for computing the minimum committed minority size required to overturn a social convention.

Heat and mass transfer analysis in flow of Walter's B nanofluid: A numerical study of dual solutions

AbstractThis study investigates the behavior of Walter's liquid B fluid over a biaxially stretching/shrinking surface for Homann stagnation point flow, focusing on the rheological properties that are crucial for understanding physical phenomena in engineering and industrial processes. The problem is modeled using Buongiorno's model in the presence of a permeable surface subject to heat generation/absorption, Ohmic heating, and chemical reactions influenced by Arrhenius activation energy. The governing partial differential equations are solved numerically using an appropriate similarity transformation, and the results are graphically analyzed via MATLAB's bvp5c solver. The study highlights the importance of characterizing the intricate properties of viscoelastic fluids, with potential applications in biomedical engineering and materials science. The findings reveal the coexistence of two distinct flow regimes and dual solutions are obtained. It is noted that the activation energy parameter is shown to enhance mass distribution in both solutions, while chemical reactions accelerate reactant conversion but reduce mass distribution. Additionally, an increase in viscoelasticity shifts the fluid's behavior towards elasticity, creating greater resistance to shear deformation and reducing both velocity profiles.

Extrinsic fluctuations in the p53 cycle.

Fluctuations are inherent to biological systems, arising from the stochastic nature of molecular interactions, and influence various aspects of system behavior, stability, and robustness. These fluctuations can be categorized as intrinsic, stemming from the system's inherent structure and dynamics, and extrinsic, arising from external factors, such as temperature variations. Understanding the interplay between these fluctuations is crucial for obtaining a comprehensive understanding of biological phenomena. However, studying these effects poses significant computational challenges. In this study, we used an underexplored methodology to analyze the effect of extrinsic fluctuations in stochastic systems using ordinary differential equations instead of solving the master equation with stochastic parameters. By incorporating temperature fluctuations into reaction rates, we explored the impact of extrinsic factors on system dynamics. We constructed a master equation and calculated the equations for the dynamics of the first two moments, offering computational efficiency compared with directly solving the chemical master equation. We applied this approach to analyze a biological oscillator, focusing on the p53 model and its response to temperature-induced extrinsic fluctuations. Our findings underscore the impact of extrinsic fluctuations on the nature of oscillations in biological systems, with alterations in oscillatory behavior depending on the characteristics of extrinsic fluctuations. We observed an increased oscillation amplitude and frequency of the p53 concentration cycle. This study provides valuable insights into the effects of extrinsic fluctuations on biological oscillations and highlights the importance of considering them in more complex systems to prevent unwanted scenarios related to health issues.

A novel fast tempered algorithm with high-accuracy scheme for 2D tempered fractional reaction-advection-subdiffusion equation

In this article, we propose a novel and computationally efficient difference scheme for evaluating the Caputo tempered fractional derivative (TFD). Our approach introduces a new fast tempered FλL2−1σ difference method, achieving a higher convergence rate of order O(Δt3−α). Specifically, we apply this method to a class of two-dimensional tempered-fractional reaction-advection-subdiffusion equations (2D-TF-RASDE) characterized by the tempered parameter λ and a tempered fractional derivative of order α, (where 0<α<1). The novel fast tempered scheme is based on the sum of exponents (SOE) technique. Additionally, we develop a novel high order compact finite difference (CFD) scheme using an alternating direction implicit (ADI) formulation for the 2D-TF-RASDE. We investigate the stability and convergence of this FLλ2−1σ-ADI-CD scheme. Numerical simulations reveal a convergence order of O(Δt1+α+hϰ4+hy4+ϵ) under robust regularity assumptions underlining the scheme's superior computational efficiency. Furthermore, our computational results align well with theoretical analysis, demonstrating both accuracy and reduced computational complexity and storage requirements as compared to the standard tempered Lλ2−1σ-ADI-CD scheme. Notably, the fast tempered FLλ2−1σ-ADI-CD scheme exhibits competitive performance and reduced CPU time relative to the standard Lλ2−1σ-ADI-CD scheme, thereby offering a distinct advantage for efficiently solving complex fractional differential equations.