Approximate Solution Research Articles

A numerical investigation of a well-known nonlinear Newell-Whitehead-Segel equation using the rank polynomial of the star graph

Mathematical models of pattern formation are indispensable tools in various fields, from developmental biology to ecology, providing insights into complex phenomena and contributing to our understanding of the natural world. These patterns have been extensively studied using reaction-diffusion and NewellWhiteheadSegel models. This article intended to find an approximate numerical solution to the NewellWhiteheadSegel equation. The appearance of stripe patterns in two-dimensional systems is explained in nonlinear systems using the NewellWhiteheadSegel equation. Based on the function basis of rank polynomials of star graphs and the well-posed operational matrices, the rank polynomial collocation method is constructed. The alleged rank polynomial collocation method created a system of nonlinear algebraic equations from the nonlinear NewellWhiteheadSegel equation. The nonlinear NewellWhiteheadSegel equation solution is approximated by solving the resulting system via Newton’s Raphson method. Numerical instances are provided to illustrate the validity and effectiveness of the technique. Verification of accuracy is accomplished by calculating error norms. The obtained numerical results show a reasonable degree of consistency with the findings reported in the current literature. The scheme’s primary benefit is the algorithm’s ease of implementation.

A handy analytical approximate solution for the magnetohydrodynamic flow of blood in a porous channel

This work presents a new version of the Picard method, known as the boundary values problems Picard method (BVPP), to obtain an analytical approximate solution for a highly complex nonlinear differential equation that models the magnetohydrodynamic flow of blood through a porous channel. The proposed method is versatile and can produce compact and easily evaluated analytical expressions that accurately capture the scientific phenomena being studied, making it ideal for practical applications. BVPP transforms a differential equation into an integral equation and utilizes an iterative algorithm like that of the basic Picard method. However, unlike the basic method, BVPP allows for the selection of an appropriate initial function and involves several adjustable parameters that can be optimized to obtain a precise analytical approximate solution with minimal effort. Overall, BVPP represents a significant advancement in the analysis of complex nonlinear differential equations, particularly in the field of biomedical engineering.

Analyzing the dynamics of a charged rotating rigid body under constant torques

This study explores the dynamical rotary motion of a charged axisymmetric spinning rigid body (RB) under the effect of a gyrostatic moment (GM). The influence of transverse and invariable body fixed torques (IBFTs), and an electromagnetic force field, is also considered. Euler’s equations of motion (EOM) are utilized to derive the regulating system of motion for the problem in a suitable formulation. Due to the lack of torque exerted along the spin axis and the nearly symmetrical nature of the RB, the spin rate is nearly unchanged. Assuming slight angular deviations of the spin axis relative to a fixed direction in space, it is possible to derive approximate analytical solutions (AS) in closed form for the attitude, translational, and rotational movements. These concise solutions that are expressed in complex form are highly effective in analyzing the maneuvers performed by spinning RBs. The study focuses on deriving the AS for various variables including angular velocities, Euler’s angles, angular momentum, transverse displacements, transverse velocities, axial displacement, and axial velocity. The graphical simulation of the subsequently obtained solutions is presented to show their precision. Furthermore, the positive impacts that alterations in the body’s parameters have on the motion’s behavior are presented graphically. The corresponding phase plane curves, highlighting the influence of different values in relation to the electromagnetic force field, the GM, and the IBFTs are drawn to analyze the stability of the body’s motion. This study has a significant role in various scientific and engineering disciplines. Its importance lies in its ability to optimize mechanical systems, explain celestial motion, and enhance spacecraft performance.

Adaptive Deep Fourier Residual method via overlapping domain decomposition

The Deep Fourier Residual (DFR) method is a specific type of variational physics-informed neural network (VPINN). It provides a robust neural network-based solution to partial differential equations (PDEs). The DFR strategy is based on minimizing the dual norm of the weak residual of a PDE, which is equivalent to minimizing the energy norm of the error. To compute the dual norm of the weak residual, the DFR method employs an orthonormal spectral basis of the test space, known for rectangles or cuboids for multiple function spaces.In this work, we extend the DFR method with ideas of traditional domain decomposition (DD). This enables two improvements: (a) to solve problems in more general polygonal domains and (b) to develop an adaptive refinement technique in the test space using a Döfler marking algorithm. In the former case, we retain the desirable equivalence between the employed loss function and the H1 error under non-restrictive assumptions, numerically demonstrating adherence to explicit bounds in the case of the L-shaped domain problem. In the latter, we show how refinement strategies lead to potentially significant improvements against a reference, classical DFR implementation with a test function space of significantly lower dimensionality, allowing us to better approximate singular solutions at a more reasonable computational cost.

A novel study of the impact of vaccination on pneumonia via fractional approach

Substantial advancements have been made in applying fractional calculus to understanding the epidemiology of viral diseases. However, it remains troubling that over 2,000,000 children, particularly those under the age of 5 and elderly individuals over 65, succumb to pneumonia annually in developing nations. This paper uses a deterministic SVEIR (susceptible, vaccinated, exposed, infected, and recovered) model to investigate pneumonia disease dynamics from a mathematical perspective. The governing model has been generalized from Caputo fractional derivatives, and the generalized Euler’s method has been used to calculate the approximate solution of the governing model. The dynamical systems theory examines the model’s stability analysis, basic reproduction number, and equilibrium points. The linearization method for spatial equilibrium and the Lyapunov functional method demonstrate that the model is locally asymptotic stable for disease-free equilibrium. Graphical results are showcased through MATLAB21 to illustrate the resolution of the problem. The research findings suggest that the disease will eventually disappear from the population if vaccination coverage rises over the necessary vaccination threshold. According to sensitivity analysis, the most sensitive factors are the transmission rate and the pace at which exposed individuals become contagious. According to the study, we must strengthen treatment effectiveness to prevent disease spreading.

Analyzing fractional effects on solutions of the generalized perturbed Zakharov-Kuznetsov equation using a residual series method with Caputo derivatives

Analyzing fractional effects on solutions of the generalized perturbed Zakharov-Kuznetsov equation using a residual series method with Caputo derivatives

Research on airport apron planning strategy in emergency situations

In events of local war or highly infectious epidemics, airports face the problem of how to park numerous non-missioned aircraft on aprons. In this study, the apron planning problem in emergency situations (APPES) was considered to replan existing parking stands and taxiways, thus determining the numbers and locations of different aircraft types. A mixed-integer programming formulation for the APPES was developed to maximise the total revenue for the apron, where the aircraft was planned to be parked based on factors such as the safety distance and aircraft taxiing method. A two-stage approach based on the discrete particle swarm optimisation algorithm and bottom-left algorithm was designed to solve the APPES. Random instances of different sizes were generated, and the experimental results demonstrated the effectiveness of the proposed approach. The two-stage approach could find optimal solutions to the APPES for 19 of the 20 small instances and could well approximate optimal solutions for the medium and large instances within a reasonable timeframe. Parking priority factor and aircraft taxiing methods sensitivity analyses were also designed to develop a strategy that can adhere to the preferences of decision makers or increase the total revenue.

Approximate analytical solutions of the 1:3 internal resonance response of piezoelectric energy harvester considering two types of coupled frequency components

The 1:3 internal resonance effect in the L-shaped piezoelectric vibration energy harvester leads to an expanded frequency band for energy harvesting and better matching of external excitation frequency in the environment. The electromechanical-coupled governing equations for this vibration energy harvester have been validated in our previous work (Nie et al., 2019). The effect of load resistance on the energy harvester’s natural frequency and damping ratio is addressed. The quadratic and cubic nonlinearities in this system leads to two types of coupled frequency components appear, they ultimately affect the approximate solution of the voltage response. The method of multiple scales is used to derive the two types of coupled frequency components of the system along with their contribution to the output responses. The revised approximate analytical solutions of the output responses are then presented. The system’s stability is assessed through the Jacobi matrix of modulation equations. The nonlinear softening phenomenon that favors the broadening of the bandwidth of the energy harvester is analyzed. The vibration patterns in regions with nonlinear softening and non-softening behavior are investigated through frequency spectra, phase trajectories, and Poincaré cross-sections. The findings reveal that the load resistance significantly affects the damping ratio of the harvester. Inclusion of quadratic and cubic nonlinearities results in three frequency components in the output responses spectrum, with an approximate ratio of 1:2:3. Two types of coupled frequency components appeared in the spectrum of the system’s output responses. The contribution of coupled frequency components to the output response proves crucial, and the revised approximate solutions are in good agreement with that of numerical solutions. The output responses are significantly affected by the initial conditions in the nonlinear softening region, the large and small initial conditions yield solutions in the upper and lower branches, respectively.

Analytic solutions of alpha-beta -time- derivatives complex finance chaotic dynamical system: synchronization and extended center manifold. An explicit approach

The present study focuses on a real finance nonlinear dynamic system (FNLDS), which has been shown to exhibit chaotic behavior. The solutions for such nonlinear dynamical systems (NLDSs) have typically been derived using numerical techniques. The objective of this study aims to; firstly, derive approximate analytical solutions for the complex FNLDS (CFNLDS) by constructing the Picard iterative scheme. The convergence of this scheme is proven, and the error analysis shows good tolerance, indicating the efficiency of the technique. Second, a novel criterion for synchronizing the real and imaginary parts of the system is presented, based on a necessary condition. Thirdly, a new method for constructing the extended center manifold is introduced. The 3D portrait reveals a feedback scroll pattern, while the 2D portrait, representing the mutual components, shows multiple pools. The synchronization of the real and imaginary parts of the system is demonstrated graphically. The FNLDS is tested for sensitivity dependence against tiny variations in the initial conditions, and it is found that the system components are moderately sensitive. Furthermore, the Hamiltonian and the extended center manifold establish a two-fold structure. It is observed that the effect of the α-β derivative leads to a delay in the behavior of the solutions.

Fractional analysis of non-linear fuzzy partial differential equations by using a direct procedure

In this study, an accurate analytical solution is presented for fuzzy FPDEs. It is done by using a novel method called the Laplace-residual power series (LRPSM) to build a series solution to the given problems. The fundamental instruments of the employed method are the Laplace transform, fractional Laurent, and fractional power series. Using the idea of a limit at infinity, we provide a series solution to a fuzzy FPDE with quick convergence and simple coefficient finding. We analyze three cases to obtain approximate and exact solutions to show the effectiveness and reliability of the Laplace- residual power series approach. To demonstrate the accuracy of the suggested procedure, we compare the findings to the real data.

Approximate solutions of the spin and pseudospin symmetries under coshine Yukawa tensor interaction

The approximate solutions of the Dirac equation for spin symmetry and pseudospin symmetry are studied with a coshine Yukawa potential model via the traditional supersymmetric approach (SUSY). To remove the degeneracies in both the spin and pseudospin symmetries, a coshine Yukawa tensor potential is proposed and applied to both the spin symmetry and the pseudospin symmetry. The proposed coshine tensor potential removes the energy degenerate doublets in both the spin symmetry and pseudospin symmetry for a very small value of the tensor strength (H = 0.05). This shows that the coshine Yukawa tensor is more effective than the real Yukawa tensor. The non-relativistic limit of the spin symmetry is obtained by using certain transformations. The results obtained showed that the coshine Yukawa potential and the real Yukawa potential has the same variation with the angular momentum number but the variation of the screening parameter with the energy for the two potential models differs. However, the energy eigenvalues of the coshine Yukawa potential model, are more bounded compared to the energies of the real Yukawa potential model.

Investigation of thermal cooling effect of long-term circulation in Raft River enhanced geothermal system

The Raft River Enhanced Geothermal System (EGS) demonstration project incorporated three hydraulic stimulations in well RRG-9 ST1 to improve geothermal production. This study investigated the injectivity index (injection rate divided by injection pressure) increase during long-term injection into RRG-9-ST1 after the stimulations from the perspective of thermoelasticity. We have used a one-dimensional approximate solution to estimate the injectivity/aperture change due to thermal cooling. The inferred aperture change vs. square root of time falls along a linear trend. The thermal coefficient (a combination of rock thermal properties, rock initial temperature, and injected fluid temperature) back-calculated by fitting the field injection data is in the same order of magnitude as the value directly calculated from these parameters. We have conducted fully-coupled thermal-hydro-mechanical modeling to simulate long-term circulation (cold water injection). The simulated injectivity index increased by 1.7 × 10−8 m3/(Pa·s) after 272 days of cold-water injection, similar to the field data, which increased by 2.5 × 10−8 m3/(Pa·s). Both the trend and the magnitude of the simulated injectivity change match with the field data recorded at Raft River. The injectivity index increased faster at the beginning and then the rate of increase became smaller with time. The injectivity index increase is hypothesized to be the result of fracture aperture increase due to cooling effects. The simulated fracture aperture at the injection well increased by 0.64 mm, and the simulated maximum fracture aperture increased by 1.93 mm. The trend of the simulated aperture change also matches the value interpreted from the field data using the 1D approximate solutions. The study suggests that cooling is the major reason for the injectivity increase at Raft River EGS during the long-term injection. More broadly, cold water injection could serve as a method to increase the injectivity in other settings after high-volume and high rate/pressure hydraulic stimulation

Semi-Analytical Closed-Form Solutions for Dynamical Ro¨ssler-Type System

Mathematical models and numerical simulations are necessary to understand the functions of biological rhythms, to comprehend the transition from simple to complex behavior and to delineate the conditions under which they arise. The aim of this work is to investigate the Ro¨ssler-type system. This system could be proposed as a theoretical model for biological rhythms, generalizing this formula for chaotic behavior. It is assumed that the Ro¨ssler-type system has a Hamilton–Poisson realization. To semi-analytically solve this system, a Bratu-type equation was explored. The approximate closed-form solutions are obtained using the Optimal Parametric Iteration Method (OPIM) using only one iteration. The advantages of this analytical procedure are reflected through a comparison between the analytical and corresponding numerical results. The obtained results are in a good agreement with the numerical results, and they highlight that our procedure is effective, accurate and usefully for implementation in applicationssuch as an oscillator with cubic and harmonic restoring forces, the Thomas–Fermi equation and the Lotka–Voltera model with three species.

An analysis of fractional piecewise derivative models of dengue transmission using deep neural network

This manuscript investigates a fractional piecewise dengue transmission model using singular and non-singular kernels. The existence results and uniqueness of the solution are established by using the approach of fixed point and in the framework of piecewise derivative and integral. To obtain the approximate solution of the considered models we apply a piecewise numerical iteration scheme which is based on Newton interpolation polynomials. Furthermore, the numerical scheme for piecewise derivatives encompasses singular and non-singular kernels. This study aims to enhance our understanding of dengue internal transmission dynamics by using a novel piecewise derivative approach that considers both singular and non-singular kernels. This work contributes to clarifying the concept of piecewise derivatives and their significance in understanding crossover dynamics. Moreover, a deep neural network approach is employed with high accuracy in training, testing, and validation of data to investigate the specified disease problem. This methodology is employed to thoroughly investigate the intricacies of the specified disease problem.

Estimation of the mass density of biological matter from refractive index measurements

The quantification of physical properties of biological matter gives rise to novel ways of understanding functional mechanisms. One of the basic biophysical properties is the mass density (MD). It affects the dynamics in sub-cellular compartments and plays a major role in defining the opto-acoustical properties of cells and tissues. As such, the MD can be connected to the refractive index (RI) via the well known Lorentz-Lorenz relation, which takes into account the polarizability of matter. However, computing the MD based on RI measurements poses a challenge, as it requires detailed knowledge of the biochemical composition of the sample. Here we propose a methodology on how to account for assumptions about the biochemical composition of the sample and respective RI measurements. To this aim, we employ the Biot mixing rule of RIs alongside the assumption of volume additivity to find an approximate relation of MD and RI. We use Monte-Carlo simulations and Gaussian propagation of uncertainty to obtain approximate analytical solutions for the respective uncertainties of MD and RI. We validate this approach by applying it to a set of well-characterized complex mixtures given by bovine milk and intralipid emulsion and employ it to estimate the MD of living zebrafish (Danio rerio) larvae trunk tissue. Our results illustrate the importance of implementing this methodology not only for MD estimations but for many other related biophysical problems, such as mechanical measurements using Brillouin microscopy and transient optical coherence elastography.

Exploring the influence of nanolayer morphology on magnetized tri-hybrid nanofluid flow using artificial neural networks and Levenberg–Marquardt optimization

This study investigates the impact of morphological nanolayers on the thermal conductivity of magnetized tri-hybrid nanofluid flow using artificial neural networks (ANNs) employing the Levenberg–Marquardt (LM-ANN) scheme. Nonlinear partial differential equations (PDEs) are transformed into nonlinear ordinary differential equations (ODEs) using similarity variables. A dataset covering various parameters such as nanolayer thickness, particle radius, magnetic parameter, expansion/contraction ratio, Schmidt number, volume fraction, and Prandtl number is generated using the Bvp5c numerical algorithm. LM-ANN is then used for testing, training, and validation to analyze approximate solutions for individual cases. Histogram analysis, mean square error (MSE), and regression investigations validate the LM-ANN. The proposed approach closely aligns with suggested and reference findings, showing an accuracy level within 10−10 range. Increasing nanolayer thickness leads to higher effective thermal conductivity. The Nusselt number increases notably with higher hybrid nanoparticle volume fraction. Laminar shape consistently exhibits superior heat transfer performance compared to other shapes. Increasing nanoparticle volume fraction enhances the Nusselt number and reduces skin friction coefficients (SFC). Industry recommendations include integrating nanolayer technologies for improved thermal efficiency.

On one solution of the problem of vibrations of mechanical systems with moving boundaries

An analytical method of solving the wave equation describing the oscillations of systems with moving boundaries is considered. By changing the variables that stop the boundaries and leave the equation invariant, the original boundary value problem is reduced to a system of functional-difference equations, which can be solved using direct and inverse methods. An inverse method is described that makes it possible to approximate quite diverse laws of boundary motion by laws obtained from solving the inverse problem. New particular solutions are obtained for a fairly wide range of laws of boundary motion. A direct asymptotic method for the approximate solution of a functional equation is considered. An estimate of the errors of the approximatemethod was made depending on the speed of the boundary movement.

Backbone Index and GNN Models for Skyline Path Query Evaluation over Multi-cost Road Networks

Skyline path queries (SPQs) extend skyline queries to multi-dimensional networks, such as multi-cost road networks (MCRNs). Such queries return a set of non-dominated paths between two given network nodes. Despite the existence of extensive works on evaluating different SPQ variants, SPQ evaluation is still very inefficient due to the nonexistence of efficient index structures to support such queries. Existing index building approaches for supporting shortest-path query execution, when directly extended to support SPQs, use an unreasonable amount of space and time to build, making them impractical for processing large graphs. In this paper, we propose a novel index structure, backbone index , and a corresponding index construction method that condenses an initial MCRN to multiple smaller summarized graphs with different granularity. We present efficient approaches to find approximate solutions to SPQs by utilizing the backbone index structure. Furthermore, considering making good use of historical query and query results, we propose two models, S kyline P ath G raph N eural N etwork (SP-GNN) and T ransfer SP-GNN (TSP-GNN), to support effective SPQ processing. Our extensive experiments on real-world large road networks show that the backbone index can support finding meaningful approximate SPQ solutions efficiently. The backbone index can be constructed in a reasonable time, which dramatically outperforms the construction of other types of indexes for road networks. As far as we know, this is the first compact index structure that can support efficient approximate SPQ evaluation on large MCRNs. The results on the SP-GNN and TSP-GNN models also show that both models can help get approximate SPQ answers efficiently.

Simulations and analysis of underwater acoustic wave propagation model based on Helmholtz equation

In this study, we explore the effectiveness of elliptic partial differential equations (PDEs) in two and three dimensional space based on Helmholtz equation for simulations of acoustic sound in a very complex environment of propagation. For this purpose, we use an advance and robust numerical technique by utilizing the properties of shifted Chebyshev spectral collocation method. This technique is an extension of the traditional Chebyshev polynomials, incorporating a shift in their argument to enhance flexibility across a wider domain, while retaining an extraordinary numerical characteristic such as orthogonality and spectral convergence making them exceptionally effective in finding the approximate solutions. The exponential order of convergence of the proposed approach is shown both through theoretical and numerical approaches. We provide a number of numerical experiments to verify the theoretical results. The spectral convergence has been substantially enhanced by these numerical examples. The exponential order is further validated by numerical error behaviour in both L 2 and L ∞ norms.

Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems

Abstract In this investigation, the fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) problem is solved using two modern semi-analytic techniques known as the Aboodh residual power series method (ARPSM) and Aboodh transform iteration method (ATIM). The two suggested approaches are briefly explained, along with how to use them to solve the fractional Hirota–Satsuma coupled KdV problem. Some analytical approximate solutions for the current problem are derived using the proposed techniques until the second-order approximation. To ensure high accuracy of the derived approximation, they are analyzed numerically and graphically and compared with the exact solutions of the integer cases. The offered techniques demonstrate more accuracy in their outcomes compared to other alternatives. The numerical results show that ARPSM and ATIM are highly accurate, practical, and beneficial for solving nonlinear equation systems. The current results are expected to help many physics researchers in modeling their different physical problems, especially those interested in plasma physics.