Method For Numerical Solution Research Articles

A numerical method for solving uncertain wave equation

Partial differential equations are usually used to deal with natural or physical problems, and wave equations are an important branch of them. Considering the fluctuation phenomena with uncertain noise in real life, this paper will focus on the study of the uncertain wave equation. The uncertain wave equation is a special kind of partial differential equation excited by the Liu process, which is widely used to describe wave propagation phenomena. However, not all uncertain wave equations can be solved analytically, so numerical solutions are of particular importance. The main objective of this paper is to study the numerical solution of uncertain wave equations. We first introduce the concept of α-path and, based on it, prove an important formula that reveals the connection between the uncertain wave equation and the classical wave equation. Based on this finding, we design a new numerical solution method to solve the uncertain wave equation. To verify the feasibility and accuracy of the method, we conduct numerical experiments. In addition, as an application, we obtain the uncertainty distribution, the inverse uncertainty distribution and the expected value of the solution of the equation based on the important formula. The originality and novelty of the conclusions of this work are demonstrated by comparing the results with those of other researchers.

Accelerated and interpretable prediction of local properties in composites

The localized stress and strain field simulation results are critical for understanding the mechanical properties of materials, such as strength and toughness. However, applying off-the-shelf machine learning or deep learning methods to a digitized microstructure restricts the image samples to be of a fixed size and also lacks interpretability. Additionally, existing methods that utilize deep learning models to solve boundary value problems require retraining the model for each set of boundary conditions. To address these limitations, we propose a customized Pixel-Wise Convolutional Neural Network (PWCNN) to make fast predictions of stress and strain fields pixel-by-pixel under different loading conditions and for a wide range of composite microstructures of any size (e.g., much larger or smaller than the sample on which the PWCNN is trained). Through numerical experiments, we show that our PWCNN model serves as an alternative approach to numerical solution methods, such as finite element analysis, but is computationally more efficient, and the prediction errors on the test microstructure are around 5%. Moreover, we also propose an interpretable machine learning framework to facilitate the scientific discovery of why certain microstructures have better or worse performance than others, which has important implications in the design of composite microstructures in advanced manufacturing.

Biyo-Hibrit Çarpışma Kutusu Tasarımının Sayısal Analizi ve Eğik Darbe Altında Enerji Sönümleme Performanslarının İncelenmesi

Designs inspired by nature are used in several areas as automotive, aerospace and defence industry. In studies on the axial crushing analysis of thin-walled structures such as crashworthiness, a critical role of developments of design aspects have been achieved by the bioinspired perspective. The energy absorption performance of the crashworthiness which shapes the basis of the crushing analysis is the main aspects of the numerical and experimental solution methods. In order to verification of finite element model, the energy absorption characteristic specifications were performed the square tube under axial loads. Using the proposed of lotus-inspired design performances of axial load condition, the newly design was prepared square tube included lotus bifurcation as called "bio-hybrid". To compare the axial and oblique loads performances of the L (lotus) and BH (bio-hybrid) structures, force-deformation curves was evaluated with characteristic properties as EA, CFE, SEA, MCF and PCF. The numerical analysis showed that the square outer frame included lotus bifurcation of crashworthiness is less suitable than natural circular lotus configuration under oblique loads. However, the lotus-inspired configuration is advantageous under axial loads

Second law analysis on cross flow of hybrid nanoliquid in a Darcy–Forchheimer medium with thermal radiative flow

This article’s objective is to examine the optimization of entropy in Darcy–Forchheimer cross-hybrid nanofluids as they flow toward a stretchy surface. There is a current because the surface is being stretched. The energy equation is broken down into its component parts including thermal radiation, heat source sink, and convective flow. In this case, copper oxide ( CuO ) and titanium dioxide ( TiO 2 ) are taken into consideration as nanoparticles, while engine oil (EO) is taken into consideration as a continuous phase fluid. In addition, we carried out an investigation into the relative performance of copper oxide ( CuO ) and titanium dioxide ( TiO 2 ) while they were suspended in water engine oil. The application of the second law of thermodynamics allows for the calculation of the rate of entropy optimization. Using a suitable transformation, nonlinear partial differential equations can be transformed into a standard system. In this work, we use the numerical built-in BVP4c solution method to generate numerical results for the derived nonlinear flow equation. Both copper oxide ( CuO ) and titanium dioxide ( TiO 2 ) undergo a graphical analysis of the effects of varying technical factors on entropy optimization, velocity, the Bejan number, and temperature. Numerical computations of the skin friction coefficient and Nusselt number for a range of fascinating parameters are performed for both nanoparticles ( TiO 2 and CuO ) . It is clear from the data that entropy optimization improves as the size of the radiation and porosity estimates reduces. The porosity parameter exhibits direct relationships to both temperature and velocity. Tabular comparisons of the current study with the previously published literature show a high degree of agreement. Copper and titanium oxide nanoparticles are used to increase Engine oil ( EO ) thermal enactment, making it a more useful base fluid. Further, some significant industrial and engineering applications are related to the present problem discourse.

Application of Müntz Orthogonal Functions on the Solution of the Fractional Bagley–Torvik Equation Using Collocation Method with Error Stimate

This paper uses Müntz orthogonal functions to numerically solve the fractional Bagley–Torvik equation with initial and boundary conditions. Müntz orthogonal functions are defined on the interval 0 , 1 and have simple and distinct real roots on this interval. For the function f ∈ L 2 0 , 1 , we obtain the best unique approximation using Müntz orthogonal functions. We obtain the Riemann–Liouville fractional integral operator for Müntz orthogonal functions so that we can reduce the complexity of calculations and increase the speed of solving the problem, which can be seen in the process of running the Maple program. To solve the fractional Bagley–Torvik equation with initial and boundary conditions, we use Müntz orthogonal functions and consider simple and distinct real roots of Müntz orthogonal functions as collocation points. By using the Riemann–Liouville fractional integral operator that we define for the Müntz orthogonal functions, the process of numerically solving the fractional Bagley–Torvik equation that is solved using Müntz orthogonal functions is reduced, and finally, we reach a system of algebraic equations. By solving algebraic equations and obtaining the vector of unknowns, the fractional Bagley–Torvik equation is solved using Müntz orthogonal functions, and the error value of the method can be calculated. The low error value of this numerical solution method shows the high accuracy of this method. With the help of the Müntz functions, we obtain the error bound for the approximation of the function. We have obtained the error bounds for the numerical method using which we solved the fractional Bagley–Torvik equation with initial and boundary conditions. Finally, we have given a numerical example to show the accuracy of the solution of the method presented in this paper. The results of solving this example using Müntz orthogonal functions and comparing the results with other methods that have been used the solve this example show the higher accuracy of the method proposed in this paper.

The Stability Of Crank – Nicholson and Explicit Methods for Numerical Solution for Sine – Gordon Equation

In This Paper, We Solve The Sine – Gordon Equation by two Numerical Methods : Crank – Nicholson and Explicit and we discuss The Stabilities , and we obtained That The Stability Crank – Nicholson Methods is More Than The Stability Of Explicit Methods.

A new meshless local integral equation method

This paper proposes a novel meshless local integral equation (LIE) method for numerical solutions of two and three-dimensional Poisson equations. The proposed method can be regarded as a new variant of the meshless local Petrov-Galerkin (MLPG) method which already has six variants, and so the proposed method can be called MLPG7. A variant of MLPG called local boundary integral equation (LBIE) method, uses a localized fundamental solution of Laplace equation as test function. This test function vanishes on local boundaries and therefore, the derived local equations do not involve the gradient of field variables. The motivation for new formulation is using an average of LBIE equation over radius of local sub-domains instead of a single sub-domain. A new kernel is introduced which is a new modification of fundamental solutions of Laplace equation. It is proved that using the new kernel as test function leads to the same local equations as averaging. Infact, the new formulation is a modification of the LBIE method for which the boundary integral is replaced with a domain integral. It is proved that new local weak formulation is equivalent with strong form. The convergence of the method for the case of regularly spaced nodal points is proved. For trial approximation, this paper uses Gaussian radial basis functions (Gaussian RBF), however, derivative free property of the method allows the use of any non-differentiable basis functions. Stability and convergence of the method are numerically studied by considering both 2D and 3D problems with regular and irregular nodal points. The method is compared with LBIE, finite differences (FD) and RBF-FD methods and the numerical results reveal the significance of the proposed method.

Mathematical modeling of malaria transmission dynamics in humans with mobility and control states

Malaria importation is one of the hypothetical drivers of malaria transmission dynamics across the globe. Several studies on malaria importation focused on the effect of the use of conventional malaria control strategies as approved by the World Health Organization (WHO) on malaria transmission dynamics but did not capture the effect of the use of traditional malaria control strategies by vigilant humans. In order to handle the aforementioned situation, a novel system of Ordinary Differential Equations (ODEs) was developed comprising the human and the malaria vector compartments. Analysis of the system was carried out to assess its quantitative properties. The novel computational algorithm used to solve the developed system of ODEs was implemented and benchmarked with the existing Runge-Kutta numerical solution method. Furthermore, simulations of different vigilant conditions useful to control malaria were carried out. The novel system of malaria models was well-posed and epidemiologically meaningful based on its quantitative properties. The novel algorithm performed relatively better in terms of model simulation accuracy than Runge-Kutta. At the best model-fit condition of 98% vigilance to the use of conventional and traditional malaria control strategies, this study revealed that malaria importation has a persistent impact on malaria transmission dynamics. In lieu of this, this study opined that total vigilance to the use of the WHO-approved and traditional malaria management tools would be the most effective control strategy against malaria importation.

The Modeling of the Rucklidge Chaotic System with Artificial Neural Networks

Chaotic systems are nonlinear systems that show sensitive dependence on initial conditions, and an immeasurably small change in initial value causes an immeasurably large change in the future state of the system. Besides, there is no randomness in chaotic systems and they have an order within themselves. Researchers use chaotic systems in many areas such as mixer systems that can make more homogeneous mixtures, encryption systems that can be used with high security, and artificial neural networks by taking the advantage of the order in this disorder. Differential equations in which chaotic systems are expressed mathematically are solved by numerical solution methods such as Heun, Euler, ODE45, RK4, RK5-Butcher and Dormand-Prince in the literature. In this research, Feed Forward Neural Network (FFNN), Layer Recurrent Neural Network (LRNN) and Cascade Forward Backpropogation Neural Network (CFNN) structures were used to model the Rucklidge chaotic system by making use of the MATLAB R2021A program Neural Network (NN) Toolbox. By comparing the results of different activation functions used in the modeling, the ANN structure that can best model the Rucklidge chaotic system has been determined. The training of the compared Artificial Neural Networks (ANNs) was carried out with the values obtained from the Euler numerical solution method, which can get satisfactory and fast results.

EXISTENCE OF SEISMIC WAVE DUE TO TALL BUILDING

In this article, we proposed and analyzed a mathematical model for the so – called city – effect in what in which an earthquake can be locally disturbed by the collective response of tall buildings in the large city. We use a set of equations c0upling vibrations buildings in buildings to motion under the ground. These equations were previously studied by different authors the existence of the seismic wave was ensured by a method of Numerical solutions. None those authors do not consider differential methods to show the existence of earthquake due to tall buildings. Furthermore, we show how we can solve for the wavenumber in a non – linear equation and we are able to find resonant frequencies coupling seismic waves and vibrating tall building which ensure the existence seismic wave due to building.

Numerical Investigation of Magnetohydrodynamic Flow of Reiner– Philippoff Nanofluid with Gyrotactic Microorganism Using Porous Medium

Nanoparticles facilitate the enrichment of heat transmission, which is crucial in many industrial and technical phenomena. The suspension of nanoparticles with microbes is another intriguing study area that is pertinent to biotechnology, health sciences, and medicinal applications. In the dispersion of nanoparticles, the conventional non-Newtonian fluid Reiner-Philippoff flows across a stretching sheet, which is examined in this article using numerical analysis. This study investigates the numerical investigation of Arrhenius reaction, heat radiation, and vicious variation variations on a Reiner-Philippoff nanofluid of MHD flow through a stretched sheet. Thus, for the current nanofluid, nanoparticles and bio-convection are highly crucial. The set of nonlinear differential equations is translated into Ordinary Differential Equations (ODEs) utilizing the requisite translation of similarities. These collected simple ODE are solved using the MATLAB computational tool bvp4c method. The graphical results for the velocity, concentration, motile microorganisms, and temperature profile are defined using the thermophoresis parameter and the Brownian motion respectively. Consider a tube containing gyrotactic microbes and a regular flow of nanofluid which is electrically conducted through a porous stretched sheet surface. This nonlinear differential problem is solved by a hybrid numerical solution method using fourth-order Runge-Kutta with shooting technique. The optimization method also performs well in terms of predicting outcomes accurately. As a result, the research applies the Bayesian Regularization Method (BRM) to improve the accuracy of the prediction results. Physical constraints are plotted against temperature, velocity, concentration, and microorganism profile trends and they are briefly described.

A Reduced Order Model Based on ANN-POD Algorithm for Steady-State Neutronics and Thermal-Hydraulics Coupling Problem

The neutronics and thermal-hydraulics (N/TH) coupling behavior analysis is a key issue for nuclear power plant design and safety analysis. Due to the high-dimensional partial differential equations (PDEs) derived from the N/TH system, it is usually time consuming to solve such a large-scale nonlinear equation by the traditional numerical solution method of PDEs. To solve this problem, this work develops a reduced order model based on the proper orthogonal decomposition (POD) and artificial neural networks (ANNs) to simulate the N/TH coupling system. In detail, the POD method is used to extract the POD modes and corresponding coefficients from a set of full-order model results under different boundary conditions. Then, the backpropagation neural network (BPNN) is utilized to map the relationship between the boundary conditions and POD coefficients. Therefore, the physical fields under the new boundary conditions could be calculated by the predicated POD coefficients from ANN and POD modes from snapshot. In order to assess the performance of an ANN-POD-based reduced order method, a simplified pressurized water reactor model under different inlet coolant temperatures and inlet coolant velocities is utilized. The results show that the new reduced order model can accurately predict the distribution of the physical fields, as well as the effective multiplication factor in the N/TH coupling nuclear system, whose relative errors are within 1%.

Thermal analysis for gas storage in salt cavern based on an improved heat transfer model

Oversimplification of the heat transfer mechanism and assumption of uniform temperature and pressure in the salt cavern could cause significant errors when modeling temperature fields. This work proposed a coupled transient flow and heat transfer model for gas storage in the underground salt cavern to reflect the thermal behaviors between gas and surrounding rock, particularly the gas Joule-Thomson effect during gas operation. Then, a fully coupled numerical solution method based on a unified matrix is presented. An average error of 3.26% was observed between the model and field data quoted in literature. The case study indicates that during the gas withdrawal period, the maximum temperature difference in the salt cavern can reach 14.94 °C, while the temperature drop of the cavity wall is only about 4.07 °C, which is quite different from the traditional assumption. A special focus was given to the gas withdrawal rate, which seemed to have the most significant influence on the evolution of the temperature field in the salt cavern. Compared with the gas withdrawal rate of 35 m3/s, reducing the gas withdrawal rate to 20 m3/s can increase the minimum temperature by approximately 23.41%. This study could add further insights into the thermal performance during gas operation in the salt cavern and help to reveal the evolution of the temperature field in the cavity and surrounding rock.

Bernoulli wavelet method for numerical solution of linear system of Fredholm integral equation of the second kind

One of the key tools for many fields of applied mathematics is the integral equations. Integral equations are widely utilized in many models, atmosphere–ocean dynamics, fluid mechanics, mathematical physics, and many other physical and engineering disciplines. A new numerical strategy based on the Bernoulli wavelet is introduced to solve system of Fredholm integral equations of second kind. In this paper, the Bernoulli wavelets are first built. Second, the system of Fredholm integral equations has been reduced into an algebraic system. In order to demonstrate the viability, and accuracy of the suggested Bernoulli wavelet approach, some numerical examples are offered at the end. The derived numerical results are examined with those from other numerical techniques and with exact solutions, demonstrating the superiority of the proposed method over those techniques. The novelty of proposed technique is that it can be extended for the numerical solution of two dimensional integral equations and differential equations appearing in engineering models, however some modifications will be required.

Cooperative Lane-Change Motion Planning for Connected and Automated Vehicle Platoons in Multi-Lane Scenarios

Multi-vehicle motion planning (MVMP) has become an emerging paradigm in connected and automated vehicles (CAVs). The cooperative lane-change movements with the coexistence of platoons and CAVs are typical scenarios on the muti-lane roads. This paper proposes an optimal control framework with the advantages of completeness and universality for platoons and CAVs’ cooperative lane-change motion planning in different task scenarios. Two typical cooperative scenarios are designed for the subsequent study of optimal modeling. The platoons’ reconfiguration and original shape maintenance are considered to reflect the universality of moving objects and the diversity of cooperative tasks. Approximately geometric contour models, dynamic externally tangent rectangle and inflated rectangle, are utilized to describe the platoon’s profile. Analytical complete collision avoidance constraints among different motion objects are constructed effectively. Other necessary constraints and the weighted cost function that minimizes lane-change time and motion energy are comprehensively considered. The optimal control models are established for the desired scenarios. Moreover, a numerical solution method combined with the simultaneous direct collocation method based on the trapezoidal rule and the barrier function method is proposed to obtain the optimal schemes. Simulation and contrast experiments are conducted for two scenarios. The results indicate that the cost function’s weight coefficients and specific lane-change tasks influence the cooperative motion planning effects and verify that the proposed optimal control framework is of reasonability, effectiveness, and unification.

Septic B-spline Collocation Method for Numerical Solution of the Generalized Nonlinear Schrodinger Equation

Bu çalışma, genelleştirilmiş lineer olmayan Schordinger (GNLS) denkleminin yüksek doğruluklu sayısal çözümünü elde etmek içindir. Çalışmada iki farklı zaman parçalanması kullanılacaktır. İlk zaman parçalanması doğruluğu iki olan ve literatürde iyi bilinen Crank-Nicolson yöntemi, ikinci zaman parçalanması ise tek adımlı ve dördüncü mertebeden doğruluğa sahip yöntemdir. Zaman parçalanması için genel bir yöntem kullanıldıktan sonra konum parçalanması için ise septik B-spline fonksiyonların kullanıldığı kolokasyon yöntemi kullanılacaktır. Zaman ve konum parçalanması uygulandıktan sonra lineer olmayan bir denklem sistemi elde edilecektir. Denklem sistemi Matlab paket programı yardımıyla çözülürken öncelikle iç iterasyonlu bir lineerleştirme kullanılacak ve istenilen zaman kadarki çözümler iteratif olarak bulunacaktır Son olarak solitary dalgasının yayılımı test problemi kullanılarak önerilen metotlar test edilmiştir.

A method for finding numerical solutions to Diophantine equations using Spiral Optimization Algorithm with Clustering (SOAC)

Diophantine equations are equations containing two or more unknowns, such that only the integer solutions are required. To find solutions of these equations numerically, we can be performed by solving an optimization problem using a metaheuristic method. In this paper, the Spiral Optimization Algorithm with Clustering (SOAC) method is proposed to find solutions to Diophantine equations in the form of polynomial, exponential, and also linear and nonlinear systems of equations. In the implementation of the method on solving some existing benchmark problems, the goal of simulation is to find all solutions only in a single run and in a short period of time. Appropriate values of required parameters are selected during the simulation. Results shows satisfactory in solving four problems in polynomial equations, four problems in exponential equations, and three problems in systems of linear and nonlinear equations. In most of cases, the results yield the same with the analytical or numerical solutions in the reference papers, and in some cases the results give more solutions.

Analysis of the Logistic Growth Model With Taylor Matrix and Collocation Method

Early analysis of infectious diseases is very important in the spread of the disease. The main aim of this study is to make important predictions and inferences for Covid 19, which is the current epidemic disease, with mathematical modeling and numerical solution methods. So we deal with the logistic growth model. We obtain carrying capacity and growth rate with Turkey epidemic data. The obtained growth rate and carrying capacity is used in the Taylor collocation method. With this method, we estimate and making predictions close to reality with Maple. We also show the estimates made with the help of graphics and tables.

Experimental modeling of Electric kinetic barrier (EKB) in Porous Medium, Explain its numerical solution methods and analysis of the relations in the Hydraulic-Electric‌ coupled flow

Experimental modeling of Electric kinetic barrier (EKB) in Porous Medium, Explain its numerical solution methods and analysis of the relations in the Hydraulic-Electric‌ coupled flow

Violation of Neumann Problem’s Solvability Condition for Poisson Equation, Involved in the Non-Linear PDEs, Containing the Reaction-Diffusion-Convection-Type Equation, at Numerical Solution by Direct Method

In this paper, we consider the 3D problem of laser-induced semiconductor plasma generation under the action of the optical pulse, which is governed by the set of coupled time-dependent non-linear PDEs involving the Poisson equation with Neumann boundary conditions. The main feature of this problem is the non-linear feedback between the Poisson equation with respect to induced electric field potential and the reaction-diffusion-convection-type equation with respect to free electron concentration and accounting for electron mobility (convection’s term). Herein, we focus on the choice of the numerical method for the Poisson equation solution with inhomogeneous Neumann boundary conditions. Despite the ubiquitous application of such a direct method as the Fast Fourier Transform for solving an elliptic problem in simple spatial domains, we demonstrate that applying a direct method for solving the problem under consideration results in a solution distortion. The reason for the Neumann problem’s solvability condition violation is the computational error’s accumulation. In contrast, applying an iterative method allows us to provide finite-difference scheme conservativeness, asymptotic stability, and high computation accuracy. For the iteration technique, we apply both an implicit alternating direction method and a new three-stage iteration process. The presented computer simulation results confirm the advantages of using iterative methods.