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Tuberculosis (TB) has a long history as a serious disease induced by its causative agent Mycobacterium tuberculosis. This pathogen manipulates the host's immune response, thereby stimulating inflammatory processes, which leads to an even greater imbalance of specific enzymes/inhibitors that contribute to tissue destruction. This work addresses a model consisting of two ordinary differential equations obtained by reducing a previously developed large-scale model describing lung damage, taking into account key metabolic pathways controlled by bacteria. The resulting system is explored as a dynamical system simulating the interaction between bio-markers (matrix metalloproteinases) of tissue destruction and the pathogen. In addition to the analysis of the mathematical model's features, we qualitatively compared the model dynamics with real clinical data and discussed their mutual correspondence. [ABSTRACT FROM AUTHOR]

Based on the Tao-Mason equation of state we have proposed a nonlinear ordinary differential equation that asymptotically converges to the compressibility factor of a pure substance or a mixture of chemical species. We have used the Dormand-Prince pair algorithm to solve the aforementioned differential equation in a purely numerical manner. Our method is devoid of the adverse convergence issues that are usually associated with the Newton-type solvers. We have provided two case studies concerning two industrially common compounds namely ethane and carbon dioxide, for the sake of exposition. For 96 points of different temperatures and pressures, our method succeeded at calculating the compressibility factor of carbon dioxide with an average absolute error of 6.53×10-5 and a maximum absolute error of 4.79×10-4. Unlike the previous root finding algorithms, we only need to perform “formal” polynomial deflations in our method, which circumvents the computation-intensive synthetic divisions, to obtain all compressibility factors offered by the Tao-Mason EOS.

We have developed a new procedure for converting nonlinear third-order ordinary differential equations into linear forms using point transformation. These linear equations are more general and easier to solve. By applying the point transformation again, we can obtain the general solutions for the original nonlinear equations. The key feature of this work is the use of examples to demonstrate the application of the derived linearization criteria to various existing problems, including third-order ordinary differential equations, secondorder ordinary differential equations under the Riccati transformation and third-order partial differential equations under travelling wave solutions. [ABSTRACT FROM AUTHOR]

Based on the Tao-Mason equation of state, we have proposed a nonlinear ordinary differential equation that asymptotically converges to the compressibility factor of a pure substance or a mixture of chemical species. We have used the Dormand-Prince pair algorithm to solve the aforementioned differential equation in a purely numerical manner. Our method is devoid of the adverse convergence issues that are usually associated with Newton-type solvers. We have provided two case studies concerning two industrially common compounds namely ethane and carbon dioxide, for the sake of exposition. For 96 points of different temperatures and pressures, our method succeeded at calculating the compressibility factor of carbon dioxide with an average absolute error of 6.53Ã--10-5 and a maximum absolute error of 4.79Ã--10-4. Unlike the previous root-finding algorithms, we only need to perform "formal" polynomial deflations in our method, which circumvents the computation-intensive synthetic divisions, to obtain all compressibility factors offered by the Tao-Mason EOS. [ABSTRACT FROM AUTHOR]

In this article, we present the Laguerre wavelet exact Parseval frame method (LWPM) for the two-dimensional flow of a rotating micropolar fluid in a porous channel with huge mass transfer. This flow is governed by highly nonlinear coupled partial differential equations (PDEs) are reduced to the nonlinear coupled ordinary differential equations (ODEs) using Berman's similarity transformation before being solved numerically by a Laguerre wavelet exact Parseval frame method. We also compared this work with the other methods in the literature available. Moreover, in the graphs of the velocity distribution and microrotation, we shown that the proposed scheme's solutions are more accurate and applicable than other existing methods in the literature. Numerical results explaining the effects of various physical parameters connected with the flow are discussed.

In this paper, the boundary layer flow of an Eyring-Powell non-Newtonian fluid over a linearly stretching sheet is solved using the combination of the quasilinearization method and the Fractional order of Rational Chebyshev function (FRC) collocation method on a semi-infinite domain. The quasilinearization method converts the equation into a sequence of linear equations then, using the FRC collocation method, these linear equations are solved. The governing nonlinear partial differential equations are reduced to the nonlinear ordinary differential equation by similarity transformations. The physical significance of the various parameters of the velocity profile is investigated through graphical figures. An accurate approximation solution is obtained and the convergence of numerical results is shown.

In the this paper, a new modified method is proposed for solving linear and nonlinear Lane-Emden type equations using first kind Chebyshev operational matrix of differentiation. The properties of first kind Chebyshev polynomial and their shifted polynomial are first presented. These properties together with the operation matrix of differentiation of first kind Chebyshev polynomial are utilized to obtain numerical solutions of a class of linear and nonlinear LaneEmden type singular initial value problems (IVPs). The absolute error of this method is graphically presented. The proposed framework is different from other numerical methods and can be used in differential equations of the same type. Several examples are illuminated to reveal the accuracy and validity of the proposed method.

The paper presents an analytical solution for the centric viscoelastic impact of two smooth balls. The contact period has two phases, compression and restitution, delimited by the moment corresponding to maximum deformation. The motion of the system is described by a nonlinear Hunt–Crossley equation that, when compared to the linear model, presents the advantage of a hysteresis loop closing in origin. There is only a single available equation obtained from the theorem of momentum. In order to solve the problem, in the literature, there are accepted different supplementary hypotheses based on energy considerations. In the present paper, the differential equation is written under a convenient form; it is shown that it can be integrated and a first integral is found—this being the main asset of the work. Then, all impact parameters can be calculated. The effect of coefficient of restitution upon all collision characteristics is emphasized, presenting importance for the compliant materials, in the domain of small coefficients of restitution. The results (variations of approach, velocity, force vs. time and hysteresis loop) are compared to two models due to Lankarani and Flores. For quasi-elastic collisions, the results are practically the same for the three models. For smaller values of the coefficient of restitution, the results of the present paper are in good agreement only to the Flores model. The simplified algorithm for the calculus of viscoelastic impact parameters is also presented. This algorithm avoids the large calculus volume required by solving the transcendental equations and definite integrals present in the mathematical model. The method proposed, based on the viscoelastic model given by Hunt and Crossley, can be extended to the elasto–visco–plastic nonlinear impact model.

In this paper, two efficient computational algorithms based on Rational and Exponential Bessel (RB and EB) functions are compared to solve several well-known classes of nonlinear Lane-Emden type models. The problems, which are define in some models of non-Newtonian fluid mechanics and mathematical physics, are nonlinear ordinary differential equations of second-order over the semi-infinite interval and have a singularity at x = 0. The nonlinear Lane-Emden equations are converted to a sequence of linear differential equations by utilizing the quasilinearization method (QLM), and then these linear equations are solved by RB and EB collocation methods. Afterwards, the obtained results are compared with the solution of other methods to demonstrate the efficiency and applicability of the proposed methods. [ABSTRACT FROM AUTHOR]

In this article, we introduce a fractional order of rational Bessel functions collocation method (FRBC) for solving the Thomas-Fermi equation. The problem is defined in the semi-infinite domain and has a singularity at $x = 0$ and its boundary condition occurs at infinity. We solve the problem on the semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration the equation is solves by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results, to show that the new method is accurate, efficient and applicable.

Highlights • The heat transfer flow of a Walters’ liquid B in a porous medium with Navier slip is investigated. • The physical problem is modelled into nonlinear PDE’s to ODE’s via similarity transformation. • The analytical solutions derived for velocity and temperature fields by Kummer’s function. • Temperature profiles are analyzed for two type of boundary heating processes namely PST and PHF. • The temperature distribution inside the stretching sheet is found to be controlled by Pr, N R and N I. Abstract This paper investigates the steady-state momentum and radiation heat transfer flow of a viscoelastic fluid in a porous media in the presence of a linear Navier slip boundary condition. The velocity of the fluid over the linear stretching sheet is varied linearly with the axial distance while a Walters’ liquid-B model is assumed for the viscosity. A similarity transformation reduces the Navier–Stokes equations to a set of partial differential equations that are converted into ordinary differential equations and solved analytically for the velocity. Moreover, heat is balanced between a temperature dependent heat source and radiation and leads to a differential equation with variable coefficients. The temperature equation is transformed to a confluent hypergeometric differential equation using the Rosseland approximation for the radiation and solved analytically. Results are discussed for two boundary conditions of the sheet, the prescribed surface temperature and the wall heat flux. Parameters like the Reynolds number, the viscoelastic parameter and the boundary slip parameter are found to determine the flow field. In addition, the Prandtl number, the radiation number, the wall temperature and the heat source/sink parameters are found to control the temperature distribution inside the stretching sheet. [ABSTRACT FROM AUTHOR]

This article deals with a class of three-point nonlinear boundary-value problems (BVPs) with Neumann type boundary conditions which arises in bridge design. The source term (nonlinear term) depends on the derivative of the solution, it is also Lipschitz continuous. We use monotone iterative technique in the presence of upper and lower solutions for both well order and reverse order case. Under some sufficient conditions we prove existence results. We also construct two examples to validate our results. These result can be used to generate a user friendly package in Mathematica or MATLAB so that solutions of nonlinear boundary-value problems can be computed.

In this paper, the nonlinear singular Thomas–Fermi differential equation for neutral atoms is solved using the fractional order of rational Chebyshev orthogonal functions (FRCs) of the first kind, F T n α ( t , L ) , on a semi-infinite domain, where L is an arbitrary numerical parameter. First, using the quasilinearization method, the equation be converted into a sequence of linear ordinary differential equations (LDEs), and then these LDEs are solved using the FRCs collocation method. Using 300 collocation points, we have obtained a very good approximation solution and the value of the initial slope y ′ ( 0 ) = − 1.5880710226113753127186845094239501095 , highly accurate to 37 decimal places. [ABSTRACT FROM AUTHOR]

In this article, we introduce a fractional order of rational Bessel functions collocation method (FRBC) for solving the Thomas-Fermi equation. The problem is defined in the semi-infinite domain and has a singularity at x = 0 and its boundary condition occurs at infinity. We solve the problem on the semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration the equation is solves by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results, to show that the new method is accurate, efficient and applicable. [ABSTRACT FROM AUTHOR]

In this paper we review several approaches to mathematical modeling of simple battery cells and develop these ideas further with emphasis on charge recovery and the response behavior of batteries to given external load. We focus on models which use few parameters and basic battery data, rather than detailed reaction and material characteristics of a specific battery cell chemistry, starting with the coupled ODE linear dynamics of the kinetic battery model. We show that a related system of PDE with Robin type boundary conditions arises in the limiting regime of a spatial kinetic battery model, and provide a new probabilistic representation of the solution in terms of Brownian motion with drift reflected at the boundaries on both sides of a finite interval. To compare linear and nonlinear dynamics in kinetic and stochastic battery models we study Markov chains with states representing available and remaining capacities of the battery. A natural scaling limit leads to a class of nonlinear ODE, which can be solved explicitly and compared with the capacities obtained for the linear models. To indicate the potential use of the modeling we discuss briefly comparison of discharge profiles and effects on battery performance. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of multiple positive solutions for the fourth-order boundary value problemu^(4)(t) = f(t,u(t)), 0 < t < 1,u(0) = u(1) = u''(0) = u''(1) = 0,where f : [0, 1] × [0,∞) → [0,∞) is continuous. Existence theorems are established via the theory of fixed point index in cones.

The Painlev'e equations are second order ordinary differential equations which can be grouped into six families, namely Painlev'e equation I, II,?, VI. This paper presents the series solution of second Painlevé equation via optimal homotopy asymptotic method (OHAM). This approach is highly efficient and it controls the convergence of the approximate solution. Comparison of the obtained solution via OHAM is provided with those obtained by Homotopy Perturbation Method (HPM), Adomian Decomposition Method (ADM), Sinc-collocation and Runge-Kutta 4 methods. It is revealed that there is an excellent agreement between OHAM and other published data which confirm the effectiveness of the OHAM. [ABSTRACT FROM AUTHOR]

The Painlev'e equations are second order ordinary differential equations which can be grouped into six families, namely Painlev'e equation I, II,…, VI. This paper presents the series solution of second Painlevé equation via optimal homotopy asymptotic method (OHAM). This approach is highly efficient and it controls the convergence of the approximate solution. Comparison of the obtained solution via OHAM is provided with those obtained by Homotopy Perturbation Method (HPM), Adomian Decomposition Method (ADM), Sinc-collocation and Runge-Kutta 4 methods. It is revealed that there is an excellent agreement between OHAM and other published data which confirm the effectiveness of the OHAM. doi:10.14456/WJST.2015.43

This paper presents a mathematical model for photo-excited carrier decay in a semiconductor. Due to the carrier trapping states and recombination centers in the bandgap, the carrier decay process is defined by the system of nonlinear differential equations. The system of nonlinear ordinary differential equations is approximated by linearized backward Euler scheme. Some a priori estimates of the discrete solution are obtained and the convergence of the linearized backward Euler method is proved. The identifiability analysis of the parameters of deep centers is performed and the fitting of the model to experimental data is done by using the genetic optimization algorithm. Results of numerical experiments are presented. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a collocation method using a new weighted orthogonal system on the half-line, namely the rational Gegenbauer functions, is introduced to solve numerically the third-order nonlinear differential equation, , where a is a constant parameter. This method solves the problems on semi-infinite domain without truncating it to a finite domain and transforming the domain of the problems to a finite domain. For , the equation is the well-known Blasius equation, which is a laminar viscous flow over a semi-infinite flat plate. We solve this equation by considering and compare the new results with the established results to show the efficiency and accuracy of the new method. [Copyright &y& Elsevier]

Abstract: A numerical technique for solving nonlinear ordinary differential equations on a semi-infinite interval is presented. We solve the Thomas–Fermi equation by the Sinc-Collocation method that converges to the solution at an exponential rate. This method is utilized to reduce the nonlinear ordinary differential equation to some algebraic equations. This method is easy to implement and yields very accurate results. [Copyright &y& Elsevier]

Abstract: In this study, we investigate the magnetohydrodynamic (MHD) viscous flow due to a shrinking sheet by employing the homotopy perturbation method (HPM) and Padé approximation. The series solution of the governing nonlinear problem is developed. Generally, the truncated series solution is adequate only in a small region when the exact solution is not reached. We overcame this limitation by using the Padé techniques, which have the advantage in turning the polynomial approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. Comparison of the present solutions is made with the results obtained by other applied methods and excellent agreement is noted. [Copyright &y& Elsevier]

Abstract: We consider the blow-up ODE problem , and a finite difference analogue, whose solution also blows up in finite time. In this paper, not only the convergence of the numerical blow-up time is proved but also the error estimate for the convergence is derived. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we propose a Lagrangian method for solving Lane–Emden equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on a Modified generalized Laguerre functions Lagrangian method. The method reduces the solution of this problem to the solution of a system of algebraic equations. We also present the comparison of this work with some well-known results and show that the present solution is acceptable. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we propose, a collocation method for solving the Blasius equation. The Blasius equation is a third-order nonlinear ordinary differential equation. This approach is based on a rational scaled generalized Laguerre function collocation method. We also present the comparison of this work with some well-known results and show that the present solution is accurate. [Copyright &y& Elsevier]