Complex Differential Equations Research Articles

A simple method for mechanical analysis of pressurized functionally graded material thick-walled cylinder

The complex function method is employed to establish a mechanical model for pressurized thick-wall cylinders made of functionally graded materials (FGM). This model is applicable to all radial varying modes of material properties, including both continuous and discontinuous variations in Young’s modulus and Poisson’s ratio. The main concept behind this mechanical model involves dividing the thick-walled cylinder into concentric thin cylinders, each equipped with a pair of analytical functions representing stress functions. By imposing continuity conditions between adjacent thin cylinders and considering the stress boundary conditions of the entire structure, all unknown forms of analytical functions can be determined. Subsequently, by establishing the correspondence relationship between these analytical functions and stress or displacement, it becomes possible to solve for the stress or displacement at any radial position within the thick-walled cylinder. Through comparison and verification against the numerical simulation results, it can demonstrate that as long as the differential scale is sufficiently small, high accuracy can be achieved in the final result. In other words, solutions obtained for multi-layered hollow cylinder converge toward those obtained for continuously graded thick-walled cylinder. Notably, by starting from the level of stress function and avoiding complex differential and integral equations, a linear equation system can provide information on stress and displacement distributions along the radial direction. Therefore, compared to other solving methods available, this proposed approach offers simplicity and applicability.

The solution of a complex integro differential equation in the right half-plane via an infinite plate weakened by a curvilinear hole

The solution of a complex integro differential equation in the right half-plane via an infinite plate weakened by a curvilinear hole

Two-phase numerical simulation of thermal and solutal transport of zero mass flux conditions over a porous deformable disc: The extension of Jeffrey-Hamel model

In recent times, numerous assertions on the thermophysical properties of nanoliquids in various flow regimes, particularly the laminar flow regimes have been documented in the literature. With this focus, this theoretical investigation is to explore multiple solutions of the laminar two-dimensional Jeffrey fluid describing the Buongiorno nanofluid model and features of heat transport near a stagnation point across a movable radial disc. Also, the study seeks to explore the effects of zero mass flux and velocity of mass transpiration embedded within a Jeffrey fluid. With the help of the ansatz similarity transformations, the study simplifies complex partial differential equations into a set of ordinary differential equations, facilitating numerical dual solutions employing the bvp4c solver. The multiple solutions become apparent in a specific range of the shrinkable parameter. The results of the present study show that the suction parameter intensifies the phenomena of friction factor in the upper solution, whereas an annulment trend is observed in the lower solution. In addition, the shear stress and heat transfer rate decline with higher impacts of the Deborah number for the first solution while a monotonical behaviour is seen for the second solution. Besides, the absolute critical value decreases owing to the change value of the Deborah number, as a result, the separation of the boundary layer accelerates. Moreover, the heat transfer rate declined by about 0.29%, 0.37%, and 0.28% for the FBS and 0.30%, 0.32%, and 0.28% for the SBS with larger values of Nta, Nba and Sca, respectively. Lastly, the outcomes of the present examination with existing data are an excellent match for the limiting cases.

Axial Green Function Method in an efficient projection scheme for incompressible Navier–Stokes flow in arbitrary domains

We introduce a new approach to solving incompressible Navier–Stokes flow. This method combines a projection scheme with the Axial Green Function Method (AGM). Based on the Kim and Moin methods, our methodology employs a predictor–corrector mechanism to achieve stable and accurate velocity corrections. Using axial Green functions, we transform complex differential equations into simpler one-dimensional integral equations. These are strategically placed along a minimal axis-parallel lines, known as axial lines, within the flow domain. This transformation makes computation and analysis more efficient from a numerical viewpoint. A significant innovation in our approach is using one-dimensional axial Green functions tailored explicitly for the reaction–diffusion ordinary differential operator. These functions efficiently handle the discrete-time derivative and viscous terms of the momentum equation. Furthermore, our approach allows for the arbitrary construction of axis-parallel lines, facilitating analysis near critical flow regions and even enabling the random distribution of these lines. We validate our proposed method through numerical examples, demonstrating the convergence of numerical solutions, the effectiveness of arbitrarily constructed axis-parallel lines, and the potential extension of our method to three-dimensional flow problems. Additionally, this study provides a robust and adaptable alternative way to solve incompressible Navier–Stokes flows, proving its effectiveness in practical applications such as Tesla valves.

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.

Modeling and Path-Following Control for Dubins Dynamics in Complex State Space

A path-following control design in the complex state space for the Dubins vehicle is proposed. The vehicle dynamics are modeled as a bilinear system of two complex ordinary differential equations with a scalar complex control input. The developed control solution represents a feedback linearizing proportional–integral–derivative servomechanism, with real valued control gains applied to complex valued tracking error components. The design is performed within the model reference control framework. It is applied to solving path-following and waypoint routing problems for curves with bounded curvature and vehicles with bounded speeds. The path-following controller regulates the system turn rate and the reference model speed. The vehicle speed dynamics are independently maintained as desired. In that regard, the developed control policy forces the system position to track a virtual target point defined by the reference model. Verifiable sufficient conditions are formulated to enable steering the Dubins vehicle along the designated routes in the presence of unknown bounded environmental disturbances, such as wind and gust. Application examples of the developed path-following control solution to waypoint routing guidance for unmanned aerial platforms are discussed.

Solution of The Duffing Equation Using Exponential Time Differencing Method

To describe the spring stiffening effect that occurs in physics and engineering problems, Georg Duffing added the cubic stiffness term to the linear harmonic oscillator equation and is now known as the Duffing oscillator. Despite its simplicity, its dynamic behavior is very diverse. In this research, the Exponential Time Difference method is introduced to solve the Duffing oscillator numerically. To formulate the ETD method, we were using the integration factors. It is a function which, when multiplied by an ordinary differential equation, produces a differential equation that can be integrated. This method is an effective numerical method for solving complex differential equations, especially equations that have strong non-linearity The ETD method delivers highly accurate numerical solutions for the Duffing oscillator, with minimal discrepancy from the analytical results. Through parameter variation, the ETD method's applicability extends to diverse Duffing oscillator configurations.

The α-path of multi-dimensional uncertain differential equations and its applications

As a complex uncertain differential equation, how to solve the multi-dimensional uncertain differential equation is a complicated and difficult problem. This paper will be devoted to the α-path of some special multi-dimensional uncertain differential equations, namely, multi-factor uncertain differential equations, nested uncertain differential equations and multi-factor nested uncertain differential equations. The α-path method is used to study the numerical solution problems of the above three special multi-dimensional uncertain differential equations. At the same time, the inverse uncertainty distributions and expected values of these three special multi-dimensional uncertain differential equations are also obtained. At last, the numerical algorithm examples are given to verify it.

A Compact Memristor Model Based on Physics-Informed Neural Networks.

Memristor devices have diverse physical models depending on their structure. In addition, the physical properties of memristors are described using complex differential equations. Therefore, it is necessary to integrate the various models of memristor into an unified physics-based model. In this paper, we propose a physics-informed neural network (PINN)-based compact memristor model. PINNs can solve complex differential equations intuitively and with ease. This methodology is used to conduct memristor physical analysis. The weight and bias extracted from the PINN are implemented in a Verilog-A circuit simulator to predict memristor device characteristics. The accuracy of the proposed model is verified using two memristor devices. The results show that PINNs can be used to extensively integrate memristor device models.

Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point

Abstract A singular nonlinear differential equation z σ d w d z = a w + z w f ( z , w ) , {z}^{\sigma }\frac{{\rm{d}}w}{{\rm{d}}z}=aw+zwf\left(z,w), where σ > 1 \sigma \gt 1 , is considered in a neighbourhood of the point z = 0 z=0 located either in the complex plane C {\mathbb{C}} if σ \sigma is a natural number, in a Riemann surface of a rational function if σ \sigma is a rational number, or in the Riemann surface of logarithmic function if σ \sigma is an irrational number. It is assumed that w = w ( z ) w=w\left(z) , a ∈ C ⧹ { 0 } a\in {\mathbb{C}}\setminus \left\{0\right\} , and that the function f f is analytic in a neighbourhood of the origin in C × C {\mathbb{C}}\times {\mathbb{C}} . Considering σ \sigma to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions w = w ( z ) w=w\left(z) in a domain that is part of a neighbourhood of the point z = 0 z=0 in C {\mathbb{C}} or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property lim z → 0 w ( z ) = 0 {\mathrm{lim}}_{z\to 0}w\left(z)=0 is proved and an asymptotic behaviour of w ( z ) w\left(z) is established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well.

Hemodynamic factors of spontaneous vertebral artery dissecting aneurysms assessed with numerical and deep learning algorithms: Role of blood pressure and asymmetry

Background and ObjectivesThe pathophysiology of spontaneous vertebral artery dissecting aneurysms (SVADA) is poorly understood. Our goal is to investigate the hemodynamic factors contributing to their formation using computational fluid dynamics (CFD) and deep learning algorithms. MethodsWe have developed software that can use patient imagery as input to recreate the vertebrobasilar arterial system, both with and without SVADA, which we used in a series of three patients. To obtain the kinematic blood flow data before and after the aneurysm forms, we utilized numerical methods to solve the complex Navier-Stokes partial differential equations. This was accomplished through the application of a finite volume solver (OpenFoam/Helyx OS). Additionally, we trained a neural ordinary differential equation (NODE) to learn and replicate the dynamical streamlines obtained from the computational fluid dynamics (CFD) simulations. ResultsIn all three cases, we observed that the equilibrium of blood pressure distributions across the VAs, at a specific vertical level, accurately predicted the future SVADA location. In the two cases where there was a dominant VA, the dissection occurred on the dominant artery where blood pressure was lower compared to the contralateral side. The SVADA sac was characterized by reduced wall shear stress (WSS) and decreased velocity magnitude related to increased turbulence. The presence of a high WSS gradient at the boundary of the SVADA may explain its extension. Streamlines generated by CFD were learned with a neural ordinary differential equation (NODE) capable of capturing the system’s dynamics to output meaningful predictions of the flow vector field upon aneurysm formation. ConclusionIn our series, asymmetry in the vertebrobasilar blood pressure distributions at and proximal to the site of the future SVADA accurately predicted its location in all patients. Deep learning algorithms can be trained to model blood flow patterns within biological systems, offering an alternative to the computationally intensive CFD. This technology has the potential to find practical applications in clinical settings.

Domain walls and vector solitons in the coupled nonlinear Schrödinger equation

We outline a program to classify domain walls (DWs) and vector solitons in the 1D two-component coupled nonlinear Schrödinger (CNLS) equation without restricting the signs or magnitudes of any coefficients. The CNLS equation is reduced first to a complex ordinary differential equation (ODE), and then to a real ODE after imposing a restriction. In the real ODE, we identify four possible equilibria including ZZ, ZN, NZ, and NN, with Z(N) denoting a zero (nonzero) value in a component, and analyze their spatial stability. We identify two types of DWs including asymmetric DWs between ZZ and NN and symmetric DWs between ZN and NZ. We identify three codimension-1 mechanisms for generating vector solitons in the real ODE including heteroclinic cycles, local bifurcations, and exact solutions. Heteroclinic cycles are formed by assembling two DWs back-to-back and generate extended bright-bright (BB), dark-dark (DD), and dark-bright (DB) solitons. Local bifurcations include the Turing (Hamiltonian–Hopf) bifurcation that generates Turing solitons with oscillatory tails and the pitchfork bifurcation that generates DB, bright-antidark, DD, and dark-antidark solitons with monotonic tails. Exact solutions include scalar bright and dark solitons with vector amplitudes. Any codimension-1 real vector soliton can be numerically continued into a codimension-0 family. Complex vector solitons have two more parameters: a dark or antidark component can be numerically continued in the wavenumber, while a bright component can be multiplied by a constant phase factor. We introduce a numerical continuation method to find real and complex vector solitons and show that DWs and DB solitons in the immiscible regime can be related by varying bifurcation parameters. We show that collisions between two DB solitons with a nonzero phase difference in their bright components typically feature a mass exchange that changes the frequencies and phases of the two bright components and the two soliton velocities.

Growth of meromorphic solutions of complex differential and difference equations

Growth of meromorphic solutions of complex differential and difference equations

On the Application of the Emad-Sara Transform Technique to the Solution of Complex Differential Equations

When attempting to express a differential equation of the first order, it is usual practice to use the equation represented by dy/dx= f(x,y). The above equation contains a function denoted by the notation embodied by f(x,y), defined on a portion of the xy-plane and dependent on two variables, x, and y, which are independent variables. The equation considered to be of the first order is the one that has just the first derivative, which is denoted by the notation dy/dx since there are no higher-order derivatives present in the equation. It is usual practice to make use of the differential equation when attempting to explain the connection that exists between a function and the derivatives of that function. Using this technique to identify functions inside a given domain in physics, chemistry, and many other fields of science is feasible. In order to do so, it is necessary to have previous knowledge about functions and the derivatives of those functions. In this article, the Emad-Sara transformation is presented as a method for identifying the general solution of complex differential equations of the first order that have constant coefficients. This transformation may be used to get the general solution of these equations. This approach, which is both practical and economical, may be used to find solutions to a wide variety of linear operator equations. These equations can range from simple to complex

Accelerating FEM-Based Corrosion Predictions Using Machine Learning

Atmospheric corrosion of metallic parts is a widespread materials degradation phenomena that is challenging to predict given its dependence on many factors (e.g. environmental, physiochemical, and part geometry). For materials with long expected service lives, accurately predicting the degree to which corrosion will degrade part performance is especially difficult due to the stochastic nature of corrosion damage spread across years or decades of service. The Finite Element Method (FEM) is a computational technique capable of providing accurate estimates of corrosion rate by numerically solving complex differential Eqs. characterizing this phenomena. Nevertheless, given the iterative nature of FEM and the computational expense required to solve these complex equations, FEM is ill-equipped for an efficient exploration of the design space to identify factors that accelerate or deter corrosion, despite its accuracy. In this work, a machine learning based surrogate model capable of providing accurate predictions of corrosion with significant computational savings is introduced. Specifically, this work leverages AdaBoosted Decision trees to provide an accurate estimate of corrosion current per width given different values of temperature, water layer thickness, molarity of the solution, and the length of the cathode for a galvanic couple of aluminum and stainless steel.

APPLICATION OF FUZZY LOGIC IN COMPUTER SYSTEMS OF MEDICAL DIAGNOSIS

The features of medical diagnostics that affect the quality of decision-making in decision support systems are considered. The purpose of this article is to briefly introduce various soft computing methodologies and present various applications in medicine. Formalized expert information on the given sets of diagnoses and their symptoms. A method has been developed for generating a diagnostic inference during therapeutic examinations based on fuzzy logic. The goal is to demonstrate the possibilities of applying soft computing to problems related to medicine. A transition is made from fuzzy values to certain physical parameters that can serve as commands to the actuator. The result of fuzzy inference, of course, will be fuzzy, but for the executive device it means absolutely nothing, therefore, to eliminate fuzziness, special mathematical methods can be applied to obtain the exact values transmitted by a number of diseases directly to the performer in medical diagnostics. Fuzzy logic and fuzzy set theory is a branch of mathematics that generalizes classical logic and set theory. The fuzzy system copes with such a task very quickly, despite the fact that instead of complex differential equations, the entire process of movement is described in terms of natural language: “norm”, “average”, “high” - as if the operator controls the system. Keywords: medical diagnostics, fuzzy logic, decision support system, fuzzy systems.

Mathematics as Determinant of Students’ HOTS Among HND Electrical and Electronic Engineering Students in Ghana

One crucial component of education is developing higher-order thinking skills (HOTS). The aim of this study is to analyze mathematics as determinant of students’ HOTS among HND electrical and electronic engineering student in Ghana. The test format tool used had two indicators, critical and creative thinking, and the subjects for the research were 488 electrical and electronic engineering students from 4 randomly selected Technical Universities in Ghana. The Cronbach Alpha reliability test was performed, and the Pearson test was used to assess the validity of the MAT instrument. Data were processed and analysed using SPSS version 26.0 software. Multiple regression was used as the estimation technique, and the results show a positive high correlation between HOTS and probability (0.757), and positive moderate correlations for algebra (0.669), functions (0.633), trigonometry and complex numbers (0.604), and calculus and differential equations (0.572). These statistics suggest that the level of understanding of mathematics concepts, particularly probability, can determine HOTS. The study's implication is that engineering mathematics curriculum developers should stress the practical applications of mathematics, especially probability in everyday life and offer opportunities for students to use their mathematical knowledge to solve real-world problems in order to develop HOTS.

Limit Directions of Julia Sets of Entire Functions and Complex Differential Equations

The limiting directions of Julia sets of infinite order entire functions are studied by combining the theory of complex dynamic system and the theory of complex differential equations, in which the lower bound of the measure of limiting direction of Julia set of entire solutions of complex differential equations is obtained.

Enhancing physics informed neural networks for solving Navier–Stokes equations

SummaryFluid mechanics is a critical field in both engineering and science. Understanding the behavior of fluids requires solving the Navier–Stokes equation (NSE). However, the NSE is a complex partial differential equation that can be challenging to solve, and classical numerical methods can be computationally expensive. In this paper, we propose enhancing physics‐informed neural networks (PINNs) by modifying the residual loss functions and incorporating new computational deep learning techniques. We present two enhanced models for solving the NSE. The first model involves developing the classical PINN for solving the NSE, based on a stream function approach to the velocity components. We have added the pressure training loss function to this model and integrated the new computational training techniques. Furthermore, we propose a second, more flexible model that directly approximates the solution of the NSE without making any assumptions. This model significantly reduces the training duration while maintaining high accuracy. Moreover, we have successfully applied this model to solve the three‐dimensional NSE. The results demonstrate the effectiveness of our approaches, offering several advantages, including high trainability, flexibility, and efficiency.

Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2

Abstract Our purpose in this article is to describe the solutions of several product-type nonlinear partial differential equations (PDEs) ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = 1 , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})=1, and ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = e g , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})={e}^{g}, where g ( z ) g\left(z) is a nonconstant polynomial and a j , b j {a}_{j},{b}_{j} , and c j ( j = 1 , 2 ) {c}_{j}\left(j=1,2) are constants in C {\mathbb{C}} . The finite-order transcendental entire solution u u of the first equation is of the following forms: u ( z 1 , z 2 ) = ± 1 a 1 a 2 + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\pm \frac{1}{\sqrt{{a}_{1}{a}_{2}}}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, or u ( z 1 , z 2 ) = 1 2 a 1 e Q ( z 1 , z 2 ) + 1 2 a 2 e − Q ( z 1 , z 2 ) + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\frac{1}{2{a}_{1}}{e}^{Q\left({z}_{1},{z}_{2})}+\frac{1}{2{a}_{2}}{e}^{-Q\left({z}_{1},{z}_{2})}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, where D = b 1 c 2 − b 2 c 1 D={b}_{1}{c}_{2}-{b}_{2}{c}_{1} , η 0 ∈ C − { 0 } {\eta }_{0}\in {\mathbb{C}}-\left\{0\right\} , and Q ( z 1 , z 2 ) = − 1 D [ ( a 1 c 2 + a 2 c 1 ) z 1 − ( a 1 b 2 + a 2 b 1 ) z 2 ] + η 1 , η 1 ∈ C . Q\left({z}_{1},{z}_{2})=-\frac{1}{D}\left[\left({a}_{1}{c}_{2}+{a}_{2}{c}_{1}){z}_{1}-\left({a}_{1}{b}_{2}+{a}_{2}{b}_{1}){z}_{2}]+{\eta }_{1},\hspace{1em}{\eta }_{1}\in {\mathbb{C}}. The description of the forms of the solutions for these PDEs demonstrates that our results are some improvements of the previous results given by Liu, Cao, and Xu [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), 227], and [K. Liu and T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ. 2013 (2013), No. 59, 1–10.]. Meantime, we list some examples to explain that the forms of solutions of our theorems are precise to some extent.