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This paper deals with the existence of forced waves for an epidemic model of West Nile virus in a shifting environment. Here a forced wave is a traveling wave with wave speed the same as the environmental shifting speed. The forced waves we constructed have the property that the waves tend to the positive endemic state of the epidemic model as the time tends to infinity. The derivation of these forced waves relies on a careful construction of a suitable lower solution with the help of Schauder’s fixed point theorem.

In this paper, we consider the second-order three point boundary value problem on time scales with integral boundary conditions on a half-line. We will use the upper and lower solution method along with the Schauder’s fixed point theorem to establish the existence of at least one solution which lies between pairs of unbounded upper and lower solutions. Further, by assuming two pairs of unbounded upper and lower solutions, the Nagumo condition on the nonlinear term involved in the first-order derivative, we will establish the existence of multiple unbounded solutions on an infinite interval by using the topological degree theory. The results of this paper extend the results of Akcan and Çetin (2018), Akcan and Hamal (2014), Eloe, Kaufmann and Tisdell (2006), and generalize the results of Lian and Geng (2011). Examples are included to illustrate the validation of the results.

This paper investigates a class of resonance boundary value problems for Hadamard-type fractional differential equations on an infinite interval. Utilizing the Leggett-Williams norm-type theorem proposed by O’Regan and Zima, the existence of positive solutions is established. The main conclusions are illustrated with an example.

This work is devoted to the study of singular strongly non-linear integro-differential equations of the type (Φ(k(t)v′(t)))′=ft,∫0tv(s)ds,v(t),v′(t),a.e.onR0+:=[0,+∞[,where f is a Carathéodory function, Φ is a strictly increasing homeomorphism, and k is a non-negative integrable function, which is allowed to vanish on a set of zero Lebesgue measure, such that 1/k∈Llocp(R0+) for a certain p>1. By considering a suitable set of assumptions, including a Nagumo–Wintner growth condition, we prove existence and non-existence results for boundary value problems associated with the non-linear integro-differential equation of our interest in the sub-critical regime on the real half line.

In this paper, we investigate the existence of a positive solution to the third-order boundary value problem where k is a positive constant, ϕ ∈ L (0;+ ∞) is nonnegative and does vanish identically on (0;+ ∞) and the function f : ℝ × (0;+ ∞) × (0;+ ∞) → ℝ is continuous and may be singular at the space variable and at its derivative.

In this paper, we investigate the existence of positive solutions for Hadamard type fractional differential system with coupled nonlocal fractional integral boundary conditions on an infinite domain. Our analysis relies on Guo-Krasnoselskii’s and Leggett-Williams fixed point theorems. The obtained results are well illustrated with the aid of examples.

In the present work, the problem of Hiemenz flow through a porous medium of a incompressible non-Newtonian Rivlin-Ericksen fluid with heat transfer is presented and newly developed analytic method, namely the homotopy analysis method (HAM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. This flow impinges normal to a plane wall with heat transfer. It has been attempted to show capabilities and wide-range applications of the homotopy analysis method in comparison with the numerical method in solving this problem. Also the convergence of the obtained HAM solution is discussed explicitly. Our reports consist of the effect of the porosity of the medium and the characteristics of the Non-Newtonian fluid on both the flow and heat.

In this paper, the nonlinear Thomas-Fermi equation for neutral atoms by using the fractional order of rational Chebyshev functions of the second kind (FRC2), (t, L), on an unbounded domain is solved, where L is an arbitrary parameter. Boyd ( ) has presented a method for calculating the optimal approximate amount of L and we have used the same method for calculating the amount of L. With the aid of quasilinearization and FRC2 collocation methods, the equation is converted to a sequence of linear algebraic equations. An excellent approximation solution of y(t), y′ (t), and y ′ (0) is obtained.

It is well known that the equation , where is a square matrix, has a unique bounded solution for any bounded continuous free term , provided the coefficient has no eigenvalues on the imaginary axis. This solution can be represented in the form The kernel is called Green’s function. In this paper, for approximate calculation of , the Newton interpolating polynomial of a special function is used. An estimate of the sensitivity of the problem is given. The results of numerical experiments are presented.

A new collocation method, namely the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) collocation method, is given for solving some nonlinear boundary value problems in the semi-infinite domain, such as equations of the unsteady isothermal flow of a gas, the third grade fluid, the Blasius, and the field equation determining the vortex profile. The method reduces the solution of the problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of the method, the numerical results of the present method are compared with several numerical results.