Tial Equations Research Articles

Electroosmotic flow and heat transfer characteristics of a class of biofluids in microchannels at high Zeta potential

Peristalsis is an important dynamic phenomenon in the field of biomedical research, and has great application prospects in microscale fluids. In recent years, this biomimetic (peristaltic) phenomenon has gained widespread attention due to its large-scale applications in various medical and industrial fields, such as radiation therapy, peristaltic blood pumps, and drug delivery systems. In this study, the electroosmotic flow and heat transfer characteristics are investigated under high wall Zeta potential and slip boundary conditions for a certain type of biological fluid that satisfies the Newtonian fluid model. Fluid flows under the joint action of external electric field, magnetic field, and Joule heating. Firstly, without using the Debye-Hückel linear approximation, the numerical solutions are given by using the Chebyshev spectral method for the nonlinear Poisson-Boltzmann equation, the fourth-order differential equation satisfied by the stream function, and the thermal energy equation. The results are compared with those obtained by using the Debye-Hückel linear approximation to demonstrate the effectiveness of the numerical method used in this study. Secondly, the effects of wall Zeta potential, Hartmann number <inline-formula><tex-math id="M11">\begin{document}$H$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M11.png"/></alternatives></inline-formula>, electroosmotic parameter <inline-formula><tex-math id="M12">\begin{document}$m$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M12.png"/></alternatives></inline-formula>, slip parameter <inline-formula><tex-math id="M13">\begin{document}$\beta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M13.png"/></alternatives></inline-formula> are discussed on the flow characteristics, peristaltic pumping, and trapping phenomena under electromagnetic environments, and the influence of Joule heating parameter <inline-formula><tex-math id="M14">\begin{document}$\gamma $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M14.png"/></alternatives></inline-formula> and Brinkman number <inline-formula><tex-math id="M15">\begin{document}$Br$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M15.png"/></alternatives></inline-formula> is explored on heat transfer characteristics. The results show that 1) wall Zeta potential plays an important role in controlling the velocity of fluid peristaltic flow; 2) the increase of electroosmotic parameter <inline-formula><tex-math id="M16">\begin{document}$m$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M16.png"/></alternatives></inline-formula> and slip parameter <inline-formula><tex-math id="M17">\begin{document}$\beta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M17.png"/></alternatives></inline-formula> increases the flow velocity in the central region of the channel, while the increase of Hartmann number <inline-formula><tex-math id="M18">\begin{document}$H$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M18.png"/></alternatives></inline-formula> hinders the flow of fluid; 3) these flow behaviors exhibit opposite trends near the channel walls; 4) the number of streamlines captured by peristaltic transport decreases with Hartmann number <inline-formula><tex-math id="M19">\begin{document}$H$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M19.png"/></alternatives></inline-formula> and electroosmotic parameter <inline-formula><tex-math id="M20">\begin{document}$m$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M20.png"/></alternatives></inline-formula> increasing; 5) the increase of Joule heating parameter <inline-formula><tex-math id="M21">\begin{document}$\gamma $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M21.png"/></alternatives></inline-formula> and Brinkman number <inline-formula><tex-math id="M22">\begin{document}$Br$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M22.png"/></alternatives></inline-formula> leads temperature to rise.

Problems on eigenvalues for ordinary diff tial equations of the second order with variable coefficients

Problems on eigenvalues for ordinary diff tial equations of the second order with variable coefficients

DIRECTLY SOLVING SECOND ORDER LINEAR BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

In this article, boundary value problems of second order linear ordinary differen- tial equations are solved directly using hybrid block one-step method. Power series of order six is adopted as basis function to derive this method through collocation and interpolation approach. Shooting method is employed to transform boundary value problem into initial value problem. Then hybrid block one step is used to approximate the solution. The perfor- mance of the new method is shown by solving some boundary value problems examined by previous methods.

AN OPTIMAL HOMOTOPY ANALYSIS METHOD BASED ON PARTICLE SWARM OPTIMIZATION: APPLICATION TO FRACTIONAL-ORDER DIFFERENTIAL EQUATION

This paper describes a new problem-solving mentality of finding optimal parameters in optimal homotopy analysis method (optimal HAM). We use particle swarm optimization (PSO) to minimize the exact square residu- al error in optimal HAM. All optimal convergence-control parameters can be found concurrently. This method can deal with optimal HAM which has fi- nite convergence-control parameters. Two nonlinear fractional-order differen- tial equations are given to illustrate the proposed algorithm. The comparison reveals that optimal HAM combined with PSO is effective and reliable. Mean- while, we give a sufficient condition for convergence of the optimal HAM for solving fractional-order equation, and try to put forward a new calculation method for the residual error.

LEAST-SQUARES SPECTRAL COLLOCATION PARALLEL METHODS FOR PARABOLIC PROBLEMS

In this paper, we study the flrst-order system least- squares (FOSLS) spectral method for parabolic partial difierential equations. There were lots of least-squares approaches to solve elliptic partial difierential equations using flnite element approxi- mation. Also, some approaches using spectral methods have been studied in recent. In order to solve the parabolic partial difieren- tial equations in parallel, we consider a parallel numerical method based on a hybrid method of the frequency-domain method and flrst-order system least-squares method. First, we transform the parabolic problem in the space-time domain to the elliptic prob- lems in the space-frequency domain. Second, we solve each elliptic problem in parallel for some frequencies using the flrst-order sys- tem least-squares method. And then we take the discrete inverse Fourier transforms in order to obtain the approximate solution in the space-time domain. We will introduce such a hybrid method and then present a numerical experiment.

Alerting Threshold Analysis of TCAS

A method for the setting and evaluation of TCAS (Traffic Alert and Collision Avoidance System) alerting threshold is proposed in this paper. Firstly, by analyzing the error and uncertainty of the flight path, a stochastic differen- tial equation is introduced to model the random aircraft motion. Then a pilot model according ICAO standards is given to simulate the TCAS alert event. And five kinds of alarm outputs are summed up by analyzing the principle of TCAS alert- ing function. Finally the evaluation system with multiple variables is established on System Operating Characteristic (SOC) curve. The validation of the Encounter model is performed under different parameters. Through the simulation un- der different threshold values, the optimal alerting threshold values are obtained. The experiments show the rationality of the TCAS threshold and verify the validity of our threshold-setting method.

Alerting Threshold Analysis of TCAS

A method for the setting and evaluation of TCAS (Traffic Alert and Collision Avoidance System) alerting threshold is proposed in this paper. Firstly, by analyzing the error and uncertainty of the flight path, a stochastic differen- tial equation is introduced to model the random aircraft motion. Then a pilot model according ICAO standards is given to simulate the TCAS alert event. And five kinds of alarm outputs are summed up by analyzing the principle of TCAS alert- ing function. Finally the evaluation system with multiple variables is established on System Operating Characteristic (SOC) curve. The validation of the Encounter model is performed under different parameters. Through the simulation un- der different threshold values, the optimal alerting threshold values are obtained. The experiments show the rationality of the TCAS threshold and verify the validity of our threshold-setting method.

On Drift Parameter Estimation in Models with Fractional Brownian Motion by Discrete Observations

We study a problem of an unknown drift parameter estimation in a stochastic differen- tial equation driven by fractional Brownian motion. We represent the likelihood ratio as a function of the observable process. The form of this representation is in general rather complicated. However, in the simplest case it can be simplified and we can discretize it to establish the a. s. convergence of the discretized version of maximum likelihood estimator to the true value of parameter. We also investigate a non-standard estimator of the drift parameter showing further its strong consistency.

NUMERICAL SOLUTION OF FRACTIONAL DELAY DIFFERENTIAL EQUATIONS USING SPLINE FUNCTIONS

In this paper, we investigate the use of suitable spline functions of polynomial form to approximate the solution of fractional delay differen- tial equations. The description of the proposed approximation method is first introduced. The error analysis and stability of the method are theoretically investigated. The proposed spline function is an extension and generalization to the polynomial spline functions used in (15). Numerical example is given to illustrate the applicability, accuracy and stability of the proposed method.

Effects of Suction and Thermal Radiation on Heat Transfer in a Third Grade Fluid over a Vertical Plate

Aims: An analysis is presented to investigate the effects ofsuction and thermal radiation on the unsteady convective flow and heat transfer in a third grade fluid over an infinite vertical plate. The plate is porous to allow for possible wall suction. Methodology: The governing time-based coupled partial differen tial equations, subjected to their boundary conditions, are solved numerically by applying an efficient and unconditionally stable Crank -Nicolson finite difference scheme. Numerical calculations are carried out for different values of dimensionless paramet ers in the problem. Results:An analysis of the results obtained establishes that the flow field is appreciably

On Existence and Uniqueness of Solution of Fuzzy Fractional Differential Equations

The purpose of this paper is to study the fuzzy fractional dieren- tial equations. We prove that fuzzy fractional dierential equation is equivalent to the fuzzy integral equation and then using this equivalence existence and uniqueness result is establish. Fuzzy derivative is consider in the Goetschel- Voxman sense and fractional derivative is consider in the Riemann Liouville sense. At the end, we give the applications of the main result.

The hyper order and fixed points of solutions of linear differential equations

In this paper, we obtain a precise estimation of the hyper order of solutions for a class of higher order linear differen- tial equation, and also investigate the exponents of convergence of the fixed points of solutions and their first derivatives for the sec- ond order case. These results generalize the results of Gundersen- Steinbart, Wittich and Chen-Shon. AMS Subject Classification: 30D35.

Implicit Shape Reconstruction of Unorganized Points Using PDE-Based Deformable 3D Manifolds

In this work we consider the problem of shape reconstruction from an unor- ganized data set which has many important applications in medical imaging, scientific computing, reverse engineering and geometric modelling. The reconstructed surface is obtained by continuously deforming an initial surface following the Partial Differen- tial Equation (PDE)-based diffusion model derived by a minimal volume-like variational formulation. The evolution is driven both by the distance from the data set and by the curvature analytically computed by it. The distance function is computed by implicit local interpolants defined in terms of radial basis functions. Space discretization of the PDE model is obtained by finite co-volume schemes and semi-implicit approach is used in time/scale. The use of a level set method for the numerical computation of the sur- face reconstruction allows us to handle complex geometry and even changing topology, without the need of user-interaction. Numerical examples demonstrate the ability of the proposed method to produce high quality reconstructions. Moreover, we show the effectiveness of the new approach to solve hole filling problems and Boolean operations between different data sets. AMS subject classifications: 65D18, 65F10, 65N08

The Fourth-Order Group Preserving Methods for the Integrations of Ordinary Differential Equations

The group-preserving schemes developed by Liu (2001) for integrat- ing ordinary differential equations system were adopted the Cayley transform and Pad´ e approximants to formulate the Lie group from its Lie algebra. However, the accuracy of those schemes is not better than second-order. In order to increase the accuracy by employing the group-preserving schemes on ordinary differen- tial equations, according to an efficient technique developed by Runge and Kutta to raise the order of accuracy from the Euler method, we combine the Runge- Kutta method on the group-preserving schemes to obtain the higher-order numeri- cal methods of group-preserving type. They provide single-step explicit time inte- grators for differential equations. Several numerical examples are examined, show- ing that the higher-order group-preserving schemes have good computational effi- ciency and high accuracy.

Applications of Prüfer transformations in the theory of ordinary differential equations

This article is a review article on the use of Prufer Transformations techniques in proving classical theorems from the theory of Ordinary Dierential Equations. We consider self-adjoint second order linear dierential equations of the form Lx = (p(t)x 0 (t)) 0 + g(t)x(t) = 0, t 2 (a,b). (?) We use Prufer transformation techniques (which are a gener- alization of Poincare phase-plane analysis) to obtain some of the main theorems of the classical theory of linear dieren- tial equations. First we prove theorems from the Oscillation Theory (Sturm Comparison theorem and Disconjugacy theo- rems). Furthermore we study the asymptotic behavior of the equation (?) when t ! 1 and we obtain necessary and suf- ficient conditions in order to have bounded solutions for (?). Finally, we consider a certain type of regular Sturm-Liouville eigenvalue problems with boundary conditions and we study their spectrum via Prufer transformations.

Universal functions and exactly solvable chaotic systems

A universal differential equation is a nontrivial differen- tial equation the solutions of which approximate to arbitrary accuracy any continuous function on any interval of the real line. On the other hand, there has been much interest in exactly solvable chaotic maps. An important problem is to generalize these results to continuos sys- tems. Theoretical analysis would allow us to prove theorems about these systems and predict new phenomena. In the present paper we discuss the concept of universal functions and their relevance to the theory of universal differential equations. We present a connection be- tween universal functions and solutions to chaotic systems. We will show the statistical independence between X(t) and X(t + �) (when � is not equal to zero) and X(t) is a solution to some chaotic systems. We will construct universal functions that behave as delta-correlated noise. We will construct universal dynamical systems with truly noisy solutions. We will discuss physically realizable dynamical systems with universal-like properties.

Numerical Methods for Simulation of Guitar Distortion Circuits

Electric guitarists prefer analog distortion effects over many digital implementations. This article suggests reasons for this and proposes that detailed study of the electrical physics of guitar distortion circuits provides insight to design more accurate emulations. This work introduces real- time emula- tion applied to guitar audio amplifi ers in the form of a tutorial about relevant numerical methods and a case study. The results here make a compelling case for simulating musical electronics using numerical methods in real time. Analog guitar distortion effect devices known as solid- state distortion boxes commonly include a diode clipper circuit with an embedded low- pass fi lter. These distortion- effect devices can be mod- eled and accurately simulated as Ordinary Differen- tial Equations (ODEs). A survey and a comparison of the basic numerical integration methods are presented as they apply to simulating circuits for audio processing, with the widely used diode clipper presented as an example. A dedicated simulator for the diode clipper has been developed to compare several numerical integration methods and their real- time feasibility. We found that implicit or semi- implicit solvers are preferred, although the prefi lter / static nonlinearity approximation comes surprisingly close to the actual solution.

Competition of Species with Intra-Specific Competition

Intra-specific competition in population dynamics can be described by integro-differen- tial equations where the integral term corresponds to nonlocal consumption of resources by in- dividuals of the same population. Already the single integro-differential equation can show the emergence of nonhomogeneous in space stationary structures and can be used to model the pro- cess of speciation, in particular, the emergence of biological species during evolution (6), (7). On the other hand, competition of two different species represents a well known and well studied model in population dynamics. In this work we study how the intra-specific competition can influ- ence the competition between species. We will prove the existence of travelling waves for the case where the support of the kernel of the integral is sufficiently narrow. Numerical simulations will be carried out in the case of large supports.

Some Progress in Conformal Geometry

This is a survey paper of our current research on the theory of partial differen- tial equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the 2-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.

Energy drift in molecular dynamics simulations

In molecular dynamics Hamiltonian systems of dieren tial equations are numerically integrated using the Stormer-Verlet method. One feature of these simulations is that there is an unphysical drift in the energy of the system over long integration periods. We study this energy drift by considering a representative system in which it can be easily observed and studied. We show that if the system is started in a random initial conguration, the error in energy of the numerically computed solution is well modeled as a continuous-time stochastic process: geometric Brownian motion. We discuss what in our model is likely to remain the same or to change if our approach is applied to more realistic molecular dynamics simulations.