Class Of Ordinary Differential Equations Research Articles

A novel control set-valued approach with application to epidemic models

This work is considered in the framework of studies dedicated to the control problems, especially in epidemiology where the scientist are concerned to develop effective control strategies to minimize the number of infected individuals. In this paper, we set this problem as an asymptotic target control problem under mixed state-control constraints, for a general class of ordinary differential equations that model the temporal evolution of disease spread. The set of initial data, from which the number of infected people decrease to zero, is generated by a new type of Lyapunov functions defined in the sense of viability theory. The associated controls are provided via selections of adequately designed feedback map. The existence of such selections is improved by using Micheal selection theorem. Finally, an application to the SIRS epidemic model, with numerical simulations, is given to show the efficiency of our approach. To the best of our knowledge, our work is the first one that used a set-valued approach based on the viability theory to deal with an epidemic control problem.

Optimal stock portfolio diversification under market constraints

The problem of optimal portfolio diversification is considered. Based on mathematical models of the dynamics of the market value formation of a single share and an optimal stock portfolio, the structure of the optimal portfolio is determined. Such models are built in a class of ordinary differential equations. One of the problems of optimal investing is optimizing the expected return of the stock portfolio for the desired level of risk. Another problem is the choice of the stock portfolio with the same expected return, but with a smaller risk. For this purpose, we use a set of acceptable and effective portfolios. This sequence of steps of the algorithm allows consistently solve two optimization problems. The problem of portfolio diversification consists of the problem of determining the moments of time and the necessity to perform such a diversification. In the article, we constructed an algorithm for determining these points of time, based on the solution of an optimal control problem. The application of this algorithm enables to select an optimal risk portfolio at a certain level of its expected profitability. It uses an efficient and acceptable set of investment portfolios.

A New Approximation Approach for Transient Differential Equation Models

Ordinary differential equation (ODE) models are frequently applied to describe the dynamics of signaling in living cells. In systems biology, ODE models are typically defined by translating relevant biochemical interactions into rate equations. The advantage of such \emph{mechanistic} models is that each dynamic variable and model parameter has its counterpart in the biological process which potentiates interpretations and enables biologically relevant conclusions. A disadvantage for such mechanistic dynamic models is, however, that they become very large in terms of the number of dynamic variables and parameters if entire cellular pathways are described. Moreover, analytical solutions of the ODEs are not available and the dynamics is nonlinear which is a challenge for numerical approaches as well as for statistically valid reasoning. Here, a complementary modeling approach based on curve fitting of a tailored \emph{retarded transient function (RTF)} is introduced which exhibits amazing capabilities in approximating ODE solutions in case of transient dynamics as it is typically observed for cellular signaling pathways. A benefit of the suggested RTF is the feasibility of self-explanatory interpretations of the parameters as response time, as amplitudes, and time constants of a transient and a sustained part of the response. Nine benchmark problems for cellular signaling were analyzed to demonstrate the suggested approach in realistic systems biology settings. The presented approach can serve as alternative modeling approach of individual time courses for large systems in the case of few observables. Moreover, it enables valuable interpretations of the response for traditional ODE models. It offers a data-driven strategy for predicting the approximate dynamic responses by an explicit function that also facilitates subsequent analytical calculations. Thus, it constitutes a promising complementary mathematical modeling strategy for situations where classical ODE modeling is cumbersome or even infeasible.

Oscillation of second order half-linear neutral differential equations with weaker restrictions on shifted arguments

Abstract Neutral differential equations are one of the most important extensions of classical ordinary differential equations and aim to give a better explanation for modeling phenomena where ordinary differential equations are insufficient. Naturally, all the questions studied in the scope of ordinary differential equations attracted the attention also for neutral differential equations. In this paper we study the oscillatory properties of second order half-linear neutral differential equations. We present oscillation criteria derived using a new approach. This approach allows us to reduce common restrictions on the deviations in arguments which are present in the currently known results of this type.

Applicable Solution for a Class of Ordinary Differential Equations with Singularity

Boundary value problems arise in many real applications such as nanofluids and other areas of applied sciences. The temperature/nanoparticles concentration are usually expressed as singular 2ndorder ODEs. So, it is a challenge to obtain the exact solution of these problems due to the difficulty of the singularity encountered in the governing equations. By means of a suitable transformation, a direct approach is introduced to solve a general class of 2nd-order ODEs. The efficiency of the obtained results is validated through selected problems in the literature. It is found that several existing solutions can be deduced as special cases of our generalized one. Moreover, the present results may be invested for similar future problems in fluid mechanics, especially nanofluids.

On the observability inequality of coupled wave equations: the case without boundary

In this paper, we study the observability and controllability of wave equations coupled by first or zero order terms on a compact manifold. We adopt the approach in Dehman-Lebeau’s paper [B. Dehman and G. Lebeau, SIAM J. Control Optim. 48 (2009) 521–550.] to prove that: the weak observability inequality holds for wave equations coupled by first order terms on compact manifold without boundary if and only if a class of ordinary differential equations related to the symbol of the first order terms along the Hamiltonian flow are exactly controllable. We also compute the higher order part of the observability constant and the observation time. By duality, we obtain the controllability of the dual control system in a finite co-dimensional space. This gives the full controllability under the assumption of unique continuation of eigenfunctions. Moreover, these results can be applied to the systems of wave equations coupled by zero order terms of cascade structure after an appropriate change of unknowns and spaces. Finally, we provide some concrete examples as applications where the unique continuation property indeed holds.

Fractional Derivatives of Some Fractional Functions and Their Applications

In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.

Global Attraction of ODE-based Mean Field Models with Hyperexponential Job Sizes

Mean field modeling is a popular approach to assess the performance of large scale computer systems. The evolution of many mean field models is characterized by a set of ordinary differential equations that have a unique fixed point. In order to prove that this unique fixed point corresponds to the limit of the stationary measures of the finite systems, the unique fixed point must be a global attractor. While global attraction was established for various systems in case of exponential job sizes, it is often unclear whether these proof techniques can be generalized to non-exponential job sizes. In this paper we show how simple monotonicity arguments can be used to prove global attraction for a broad class of ordinary differential equations that capture the evolution of mean field models with hyperexponential job sizes. This class includes both existing as well as previously unstudied load balancing schemes and can be used for systems with either finite or infinite buffers. The main novelty of the approach exists in using a Coxian representation for the hyperexponential job sizes and a partial order that is stronger than the componentwise partial order used in the exponential case.

Generalized RK integrators for solving class of sixth-order ordinary differential equations

In this paper, the classical RK integrators for solving quasi-linear(QL) ordinary differential equations (ODEs) up to 5th-order have been generalized to solve class of 6th-order ODEs. The novel contribution of this work is the derivation of two efficient direct integrators for solving class of ODEs. Non-linear equations of order conditions (OCs) for the proposed RKM methods have derived up to the 7th-order using Taylor expansion. Two RKM integrators of three-stages, 5th-order and four-stages, 6th-order have been derived depend on these OCs. The proposed integrates have been tested using some implementations in order to compare them with classical methods which show that the proposed integrators are less than classical RK methods in term of time complexity and function evaluations.

On solutions of the combined KdV-nKdV equation

The aim of this work is to deal with a new integrable nonlinear equation of wave propagation, the combined of the Korteweg-de vries equation and the negative order Korteweg-de vries equation (combined KdV-nKdV) equation, which was more recently proposed by Wazwaz. Upon using wave reduction variable, it turns out that the reduced combined KdV-nKdV equation is alike the reduced (3+1)-dimensional Jimbo Miwa (JM) equation, the reduced (3+1)-dimensional Potential Yu-Toda-Sasa-Fukuyama (PYTSF) equation and the reduced (3 + 1)¬dimensional generalized shallow water (GSW) equation in the trav¬elling wave. In fact, the four transformed equations belong to the same class of ordinary differential equation. With the benefit of a well known general solutions for the reduced equation, we show that sub¬jects to some scaling and change of parameters, a variety of families of solutions are constructed for the combined KdV-nKdV equation which can be expressed in terms of rational functions, exponential functions and periodic solutions of trigonometric functions and hyperbolic func¬tions. In addition to that the equation admits solitary waves, and double periodic waves in terms of special functions such as Jacobian elliptic functions and Weierstrass elliptic functions.

Global Attraction of ODE-based Mean Field Models with Hyperexponential Job Sizes

Mean field modeling is a popular approach to assess the performance of large scale computer systems. The evolution of many mean field models is characterized by a set of ordinary differential equations that have a unique fixed point. In order to prove that this unique fixed point corresponds to the limit of the stationary measures of the finite systems, the unique fixed point must be a global attractor. While global attraction was established for various systems in case of exponential job sizes, it is often unclear whether these proof techniques can be generalized to non-exponential job sizes. In this paper we show how simple monotonicity arguments can be used to prove global attraction for a broad class of ordinary differential equations that capture the evolution of mean field models with hyperexponential job sizes. This class includes both existing as well as previously unstudied load balancing schemes and can be used for systems with either finite or infinite buffers. The main novelty of the approach exists in using a Coxian representation for the hyperexponential job sizes and a partial order that is stronger than the componentwise partial order used in the exponential case.

On rapidly growing solutions of a class of ordinary differential equations

For high-order ordinary differential equations containing terms with lower order derivatives, sufficient conditions for the absence of rapidly growing solutions are obtained.

On a generalized non-local two-temperature heat transfer DAE modeling/simulation methodology for metal-nonmetal thermal inter-facial problems

With the increasing demand of electronic devices at micro-/nano-scales, the numerical modeling of metal-nonmetal thermal inter-facial problems is one of the most important and necessary aspects for the design of these devices at small scales; particularly the non-local and non-Fourier thermal behaviors play a vital role. This work exploits a generalized two-temperature heat transfer model to describe the metal-nonmetal thermal contact, in which the electron-phonon interaction dominates the thermal behavior in the metal whereas only phonon transport dominates the thermal behaviors in the nonmetal; and the heat energy between the metal and the nonmetal domains are transferred only by phonons. The non-locality in space and time of this generalized two-temperature theory is first mathematically proved with rigor. A generalized computational strategy based on this non-local two-temperature model is then formulated in the framework of differential algebraic equations instead of classical ordinary differential equations, where the thermal interface conditions are treated as algebraic equations. The proposed formulation allows considering the complex thermal inter-facial phenomena involving a large range of scale effects and Fourier/nonFourier thermal behaviors in both the metal and the nonmetal domains. The novel unified time integration of GS4-1 DAE Index 2 is exploited to solve the resulting differential-algebraic system and to predict the thermal behaviors between many thermal flux models without extra computational cost.

On Rapidly Growing Solutions of a Class of Ordinary Differential Equations

For high-order ordinary differential equations containing terms with lower order derivatives, sufficient conditions for the absence of rapidly growing solutions are obtained.

Iterative solutions to classical second-order ordinary differential equations

Iterative solutions to classical second-order ordinary differential equations

Prospective teachers' difficulties in second order linear differential equation: a case of constructing methods in solving a non-homogeneous problem

This study aims to explore students’ difficulty in constructing a method used to solve non-homogeneous problems in 2nd order ordinary differential equations. The research method used is descriptive-explorative research. The data collected were in the form of responses of 46 prospective teachers of the class of ordinary differential equations semester 6 studying at Universitas Negeri Surabaya in Indonesia. Data obtained using a student work sheet in which consist of several steps in order to construct a method can be used to find the solution of non-homogeneous in 2nd order ordinary differential equations. The results show that prospective teachers have some difficulties in making relations between two or more equations they had found. We found that 69.5% of students had most difficulty in emerging an equation consists of u1′(t) and u2′(t) so that they didn’t really understand the concept of solutions of equations.

Classification of Certain Class of Ordinary Differential Equations of the First Order

We study problem of global classification of ordinary differential equations with the linear-fractional right-hand side with rational coefficients with respect to a symmetry group. We find the field of differential invariants and obtain the equivalence criterion for two such equations. We adduce certain examples for applying of this criterion. These examples were obtained by means of computer.

On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative

In this paper, we discuss the conditions of existence and uniqueness of solutions to a certain class of ordinary differential equations involving Atangana–Baleanu fractional derivative. Benefiting from the Gronwall inequality in the frame of Riemann–Liouville fractional integral, we establish a Gronwall inequality in the frame of Atangana–Baleanu fractional integral. Then, we study the stability of such equations in the sense of Ulam.

Robust data-driven discovery of governing physical laws with error bars.

Discovering governing physical laws from noisy data is a grand challenge in many science and engineering research areas. We present a new approach to data-driven discovery of ordinary differential equations (ODEs) and partial differential equations (PDEs), in explicit or implicit form. We demonstrate our approach on a wide range of problems, including shallow water equations and Navier-Stokes equations. The key idea is to select candidate terms for the underlying equations using dimensional analysis, and to approximate the weights of the terms with error bars using our threshold sparse Bayesian regression. This new algorithm employs Bayesian inference to tune the hyperparameters automatically. Our approach is effective, robust and able to quantify uncertainties by providing an error bar for each discovered candidate equation. The effectiveness of our algorithm is demonstrated through a collection of classical ODEs and PDEs. Numerical experiments demonstrate the robustness of our algorithm with respect to noisy data and its ability to discover various candidate equations with error bars that represent the quantified uncertainties. Detailed comparisons with the sequential threshold least-squares algorithm and the lasso algorithm are studied from noisy time-series measurements and indicate that the proposed method provides more robust and accurate results. In addition, the data-driven prediction of dynamics with error bars using discovered governing physical laws is more accurate and robust than classical polynomial regressions.

Sine-Gordon expansion method for exact solutions to conformable time fractional equations in RLW-class

The Sine-Gordon expansion method is implemented to construct exact solutions some conformable time fractional equations in Regularized Long Wave (RLW)-class. Compatible wave transform reduces the governing equation to classical ordinary differential equation. The homogeneous balance procedure gives the order of the predicted polynomial-type solution that is inspired from well-known Sine-Gordon equation. The substitution of this solution follows the previous step. Equating the coefficients of the powers of predicted solution leads a system of algebraic equations. The solution of resultant system for coefficients gives the necessary relations among the parameters and the coefficients to be able construct the solutions. Some solutions are simulated for some particular choices of parameters.