Types Of Differential Equations Research Articles

An Optimal Tropospheric Tomography Method Based on the Multi-GNSS Observations

Aside from the well-known applications (positioning, navigation and timing) brought by Global Navigation Satellite System (GNSS), reconstruction of tropospheric atmosphere distribution information using tomography technique based on the multi-GNSS observations has been developed as a research point in the fields of GNSS Meteorology. In this paper, an optimal tropospheric tomography method using observations from multi-GNSS (Global Navigation Satellite System) is proposed, which considers the reasonable weightings of observation equations derived from multi-GNSS as well as the various constraints. Comparing to the equal weighting strategy of multi-GNSS observations for the previously multi-GNSS tomography studies, the proposed method in this paper has the ability to tune the weightings for a different type of equations. Experiments show that the proposed method can improve the internal/external accuracy of GNSS tomography modeling with the GNSS precise point positioning (PPP)-estimated slant wet delay as reference when compared to the conventional method. In addition, the data derived from radiosonde is used as an external testing, and the result also expresses the superiority of the proposed method when compared to the conventional method.

Exact harmonic solution to ballistic type heat propagation in thin films and wires

The system of hyperbolic type differential equations is considered for non–Fourier heat transfer in thin films and wires. Structures of the harmonic solutions for telegrapher equation and for Guyer–Krumhansl equation are identified and compared with each other. The exact harmonic solution is obtained; the influence of the phonon heat transfer is explored. The maximum principle violation is demonstrated. Frequency dependence of the solution for Guyer–Krumhansl equation is shown. The exact analytical solution is derived for the inhomogeneous ballistic–diffusion set of differential equations, governing heat propagation in thin films. The contribution of the ballistic, wave and diffusive heat transfer components as well as the influence of the initial conditions and of the Knudsen number on heat conduction in thin films and wires are investigated. The cases of identical and different from each other Knudsen numbers for ballistic and diffusive components are explored. The examples of exact solutions for the heat transfer in each case are demonstrated.

Homoclinic and stable periodic solutions for differential delay equations from physiology

A one-parameter family of Mackey-Glass type differential delay equations is considered. The existence of a homoclinic solution for suitable parameter value is proved. As a consequence, one obtains stable periodic solutions for nearby parameter values. An example of a nonlinear functions is given, for which all sufficient conditions of our theoretical results can be verified numerically. Numerically computed solutions are shown.

Demand Functions in Dynamic Games

Abstract The paper is devoted to construction of solutions in dynamic bimatrix games. In the model, the payoffs are presented by discounted integrals on the infinite time horizon. The dynamics of the game is subject to the system of the A.N. Kolmogorov type differential equations. The problem of construction of equilibrium trajectories is analyzed in the framework of the minimax approach proposed by N.N. Krasovskii and A.I. Subbotin in the differential games theory. The concept of dynamic Nash equilibrium developed by A.F. Kleimenov is applied to design the structure of the game solution. For obtaining constructive control strategies of players, the maximum principle of L.S. Pontryagin is used in conjunction with the generalized method of characteristics for Hamilton-Jacobi equations. The impact of the discount index is indicated for equilibrium strategies of the game and demand functions in the dynamic bimatrix game are constructed.

The combined Laplace transform and new homotopy perturbation methods for Lane-Emden Type Differential Equations

The combined Laplace transform and new homotopy perturbation methods for Lane-Emden Type Differential Equations

A troposphere tomography method considering the weighting of input information

Abstract. Troposphere tomography measurement using a global navigation satellite system (GNSS) generally consists of several types of input information including the observation equation, horizontal constraint equation, vertical constraint equation, and a priori constraint equation. The reasonable weightings of input information are a prerequisite for ensuring the reliability of the adjustment of the parameters. This forms the focus of this research, which tries to determine the weightings, including the observations, for the same type of equation and the optimal weightings for different types of equations. The optimal weightings of the proposed method are realized on the basis of the stable equilibrium relationship between different types of a posteriori unit weight variances, which are capable of adaptively adjusting the weightings for different types of equations and enables the ratio between the two arbitrary a posteriori unit weight variances to tend to unity. A troposphere tomography experiment, which was used to consider these weightings, was implemented using global positioning system (GPS) data from the Hong Kong Satellite Positioning Reference Station Network (SatRef). Numerical results show the applicability and stability of the proposed method for GPS troposphere tomography assessment under different weather conditions. In addition, the root mean square (RMS) error in the water vapor density differences between tomography-radiosonde and tomography-ECMWF (European Centre for Medium-Range Weather Forecasts) are 0.91 and 1.63 g m−3, respectively, over a 21-day test.

Numerical Integration Approximation of Integral over a given Interval

Numerical integration is used to approximate an integral over a certain interval. The purpose of the numerical integration is that it is not always possible to evaluate the integral analytically. So, the numerical integration is used. This articles deals with the approximation of the integral based on the Jacobi and the Romanovski polynomials. This is a quadrature for evaluating the integral. Jacobi and Romanovski polynomials are seen as advanced polynomials that are a solution of a certain Sturm-Liouville type differential equations which is described in this paper.

Comments on the properties of Mittag-Leffler function

The properties of Mittag-Leffler function is reviewed within the framework of an umbral formalism. We take advantage from the formal equivalence with the exponential function to define the relevant semigroup properties. We analyse the relevant role in the solution of Schr\"odinger type and heat-type fractional partial differential equations and explore the problem of operatorial ordering finding appropriate rules when non-commuting operators are involved. We discuss the coherent states associated with the fractional Sch\"odinger equation, analyze the relevant Poisson type probability amplitude and compare with analogous results already obtained in the literature.

Optimization of Quantitative Dynamic Postgadolinium MRI Technique Using Normalized Ratios for the Evaluation of Temporomandibular Joint Synovitis in Patients with Juvenile Idiopathic Arthritis.

MR imaging has been shown to be useful in the diagnosis of juvenile idiopathic arthritis of the temporomandibular joint. Prior MR imaging approaches have relied mainly on the subjective interpretation of synovial enhancement as a marker for synovial inflammation. Although, more recently, several attempts have been made to quantify synovial enhancement, these methods have not taken into account the dynamic enhancement characteristics of the temporomandibular joint and the effect of sampling time. Our aim was to develop a clinically feasible, reproducible, dynamic, contrast-enhanced MR imaging technique for the quantitative assessment of temporomandibular joint synovitis in patients with juvenile idiopathic arthritis and to study the effect of sampling time on the evaluation of synovitis. This was a retrospective study of all patients who had dynamic, contrast-enhanced coronal T1 3T MR imaging through the temporomandibular joint at our institution between January 1, 2015, and July 8, 2016. Patients in this cohort included those with a history of juvenile idiopathic arthritis and control patients who underwent MR imaging for other routine, clinical purposes. Synovial enhancement was calculated for each temporomandibular joint using 3 different types of equations termed normalization ratios. The enhancement profiles generated by each equation were studied to determine which provided the best discrimination between affected and unaffected joints, was the least susceptible to sampling errors, and was the most clinically feasible. A ratio of synovial enhancement (defined as the difference between the postgadolinium and the pregadolinium T1 signal of the synovium) to the postgadolinium signal of the longus capitis provided the best discrimination between affected and unaffected joints, the least susceptibility to sampling error, and was thought to be the most clinically feasible method of quantification of synovial inflammation. Additional synovial enhancement ratios studied did not provide the same level rates of discrimination between the affected and unaffected joints and were thought to be too temporally variable to provide reliable clinical use. We provide a robust, reproducible, dynamic gadolinium-enhanced MR imaging technique for the quantitative assessment of temporomandibular joint synovitis in patients with juvenile idiopathic arthritis.

Coupling of finite element method and discontinuous Galerkin method to simulate viscoelastic flows

SummaryIn this paper, a numerical method, which is about the coupling of continuous and discontinuous Galerkin method based on the splitting scheme, is presented for the calculation of viscoelastic flows of the Oldroyd‐B fluid. The momentum equation is discretized in time by using the Adams‐Bashforth second‐order algorithm, and then decoupled via the splitting approach. Considering the Oldroyd‐B constitutive equation, the second‐order Runge‐Kutta approach is selected to complete the temporal discretization. As for the spatial discretizations, the fundamental purpose is to make the best of finite element method (FEM) and discontinuous Galerkin (DG) method to handle different types of equations. Specifically speaking, for the subequations, FEM is chosen to treat the Poisson and Helmholtz equations, and DG is employed to deal with the nonlinear convective term. In addition, because of the hyperbolic nature, DG is also utilized to discretize the Oldroyd‐B constitutive equation spatially. This coupled method avoids resorting to extra stabilization technique occurred in standard FEM framework even for moderately high values of Weissenberg number and also reduces the complexity compared with unified DG scheme. The Oldroyd‐B model is applied to investigate several typical and challenging benchmarks, such as the 4:1 planar contraction flow and the lid‐driven cavity flow, with a wide range of Weissenberg number to illustrate the feasibility, robustness, and validity of our coupled method.

Determination of the excitation frequencies of laminated orthotropic non-homogeneous conical shells

Abstract The excitation frequencies of parametric vibration of laminated non-homogeneous orthotropic conical shells (LNHOCSs) under axial load periodically varying with time, are determined using the classical shell theory (CST). The basic equations are found using the Donnell-Mushtari shell theory and reduce to the Mathieu-Hill type differential equation, in which the instability is examined by the Bolotin method. To validate of current results was made a comparison with the previous studies. The effects of stacking sequences, axial load factors, non-homogeneity, as well as the variation of geometric characteristics on the backward and forward excitation frequencies (BFEFs) of conical shells are studied in detail.

Manton’s five vortex equations from self-duality

We demonstrate that the five vortex equations recently introduced by Manton arise as symmetry reductions of the anti-self-dual Yang–Mills equations in four dimensions. In particular the Jackiw–Pi vortex and the Ambjørn–Olesen vortex correspond to the gauge group , and respectively the Euclidean or the symmetry groups acting with two-dimensional orbits. We show how to obtain vortices with higher vortex numbers, by superposing vortex equations of different types. Finally we use the kinetic energy of the Yang–Mills theory in 4 + 1 dimensions to construct a metric on vortex moduli spaces. This metric is not positive-definite in cases of non-compact gauge groups.

Harnack inequality for nonlinear elliptic equations with strong absorption

We develop Harnack inequalities for two different types of equations. First we consider a fully nonlinear uniformly elliptic equation related to the Pucci's maximal and the minimal operators. Next we consider a quasilinear equation related to the p-Laplacian. In both cases we consider lower order terms of Keller–Osserman type. Although the equations considered are quite different, we employ a unified method to approach both problems and the results we find are similar.

Data-driven non-linear elasticity: constitutive manifold construction and problem discretization

The use of constitutive equations calibrated from data has been implemented into standard numerical solvers for successfully addressing a variety problems encountered in simulation-based engineering sciences (SBES). However, the complexity remains constantly increasing due to the need of increasingly detailed models as well as the use of engineered materials. Data-Driven simulation constitutes a potential change of paradigm in SBES. Standard simulation in computational mechanics is based on the use of two very different types of equations. The first one, of axiomatic character, is related to balance laws (momentum, mass, energy, $$\ldots $$ ), whereas the second one consists of models that scientists have extracted from collected, either natural or synthetic, data. Data-driven (or data-intensive) simulation consists of directly linking experimental data to computers in order to perform numerical simulations. These simulations will employ laws, universally recognized as epistemic, while minimizing the need of explicit, often phenomenological, models. The main drawback of such an approach is the large amount of required data, some of them inaccessible from the nowadays testing facilities. Such difficulty can be circumvented in many cases, and in any case alleviated, by considering complex tests, collecting as many data as possible and then using a data-driven inverse approach in order to generate the whole constitutive manifold from few complex experimental tests, as discussed in the present work.

On Kiguradze’s problem on power-law asymptotic behavior of blow-up solutions to Emden–Fowler type differential equations

Abstract For higher-order Emden–Fowler type equations with regular nonlinearity, the asymptotic behavior of their blow-up solutions is investigated. It is proved that for weakly superlinear differential equations, all such solutions have power-law asymptotic behavior.

Analytical approximate solutions for quadratic Riccati differential equation of fractional order using the Polynomial Least Squares Method

In this paper the Polynomial Least Squares Method is applied in order to compute analytical approximate polynomial solutions for the fractional Riccati type differential equations. The accuracy of the method is tested by means of the comparison with previous results for several applications.

Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients

Abstract In this paper, we compute explicitly the oscillation constant for certain half-linear second-order differential equations having different periodic coefficients. Our result covers known result concerning half-linear Euler type differential equations with α—periodic positive coefficients. Additionally, our result is new and original in case that the least common multiple of these periods is not defined. We give an example and corollaries which illustrate cases that are solved with our result.

Existence of solutions for a mixed type differential equation with state-dependence

In this paper we study a general n-dimensional mixed type differential equation with state dependence. Many known works give the existence of solutions with the so-called Return Condition. In this paper, without requiring the Return Condition, we prove a non-local existence of solutions by using a technique of compactification and the dependence of the maximum of functions on the size of their domains. We apply our result to differential equations with iterates.

Analysis of a diffuse interface model of multispecies tumor growth

We consider a diffuse interface model for tumor growth recently proposed in Chen et al (2014 Int. J. Numer. Methods Biomed. Eng. 30 726–54). In this new approach sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species. Hence, a continuum thermodynamically consistent model is introduced. The resulting PDE system couples four different types of equations: a Cahn–Hilliard type equation for the tumor cells (which include proliferating and dead cells), a Darcy law for the tissue velocity field, whose divergence may be different from 0 and depend on the other variables, a transport equation for the proliferating (viable) tumor cells, and a quasi-static reaction diffusion equation for the nutrient concentration. We establish existence of weak solutions for the PDE system coupled with suitable initial and boundary conditions. In particular, the proliferation function at the boundary is supposed to be nonnegative on the set where the velocity satisfies , where ν is the outer normal to the boundary of the domain.

The effect of heterogeneity on the parametric instability of axially excited orthotropic conical shells

The parametric instability of heterogeneous orthotropic truncated conical shells under time dependent axial compressive load on the basis of Donnell shell theory is investigated. The dynamic basic equations are reduced into a Mathieu-Hill type differential equation describing the instability of heterogeneous orthotropic truncated conical shells using Galerkin's method. The backward and forward excitation frequencies are determined by using Bolotin's method. A comparison with the previous studies has been developed in order to validate the present results. The effects of axial load factors, heterogeneity, orthotropy, as well as the variation of the characteristics of the conical shell on the backward and forward excitation frequencies are studied in detail.