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In this paper, we consider the inverse scattering problem for Sturm-Liouville operator with discontinuity conditions at some point on the positive half line. The scattering data of this boundary value problem is examined. The resolvent operator is constructed and the expansion formula with respect to the eigenfunctions of this boundary value problem is obtained. The main equation or modified Marchenko equation of the inverse scattering problem is derived and an algorithm of the construction of the potential function according to scattering data of this boundary value problem is given.

Consider the advective Fisher-KPP equation u t = u x x − β u x + f ( u ) on the half line [ 0 , ∞ ) with Dirichlet boundary condition at x = 0 . In a recent paper [10] , the authors considered the problem without advection (i.e., β = 0 ) and constructed a new type of entire solution U ( x , t ) , which, under the additional assumption f ″ ( u ) ≤ 0 , is concave and U ( ∞ , t ) = 1 for all t ∈ R . In this paper, we consider the equation with advection and without the additional assumption f ″ ( u ) ≤ 0 . In case β = 0 , using a quite different approach from [10] we construct an entire solution U ˜ which is similar as U in the sense that U ˜ ( ∞ , t ) ≡ 1 and U ˜ ( ⋅ , t ) is asymptotically flat as t → − ∞ , but different from U in the sense that it does not have to be concave. Our result reveals that the asymptotically flat (as t → − ∞ ) property rather than the concavity is more essential for such entire solutions. In case β 0 , we construct another new entire solution U ˆ which is completely different from the previous ones in the sense that U ˆ ( ∞ , t ) increases from 0 to 1 as t increasing from −∞ to ∞.

The purpose of this paper is to analyze a new kind of Hadamard fractional boundary value problem combining integral boundary condition and multipoint fractional integral boundary condition on an infinite interval. By the help of the Bai-Ge’s fixed point theorem, multiplicity results of positive solutions are derived for the Hadamard fractional boundary value problem. In the end, to illustrative the main result, an example is also presented.

In this paper, we improve the existing results in the literature by presenting weaker sufficient conditions for the solvability of a third-order impulsive problem on the half-line, having generalized impulse effects. More precisely, our nonlinearities do not need to be positive nor sublinear and the monotone assumptions are local ones. Our method makes use of some truncation and perturbed techniques and on the equiconvergence at infinity and the impulsive points. The last section contains an application to a boundary layer flow problem over a stretching sheet with and without heat transfer.

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

In this paper, we investigate the existence of a positive solution to the third-order boundary value problem { - u ‴ ( t ) + k 2 u ′ ( t ) = φ ( t ) f ( t , u ( t ) , u ′ ( t ) ) , t > 0 u ( 0 ) = u ′ ( 0 ) = u ′ ( + ∞ ) = 0 , \left\{ \matrix{- u'''\left( t \right) + {k^2}u'\left( t \right) = \phi \left( t \right)f\left( {t,u\left( t \right),u'\left( t \right)} \right),\,\,\,t > 0 \hfill \cr u\left( 0 \right) = u'\left( 0 \right) = u'\left( { + \infty } \right) = 0, \hfill \cr} \right. where k is a positive constant, ϕ ∈ L1 (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.

This paper is devoted to the existence of solutions for a class of nonlinear boundary value problems with integral boundary conditions and generalized 𝑝-Laplacian on the positive half-line. We establish sufficient conditions to guarantee the existence of solutions in a special function space by using Leray–Schauder-type arguments. Examples are also given to illustrate the main results.

In this paper, we consider Galerkin and multi-Galerkin methods and their iterated versions for solving the nonlinear Hammerstein-type integral equation on the half-line with sufficiently smooth ker...

The aim of the present paper is to study the existence of nontrivial nonnegative solutions for a second-order boundary value problem of Kirchhoff type on the half-line. Our approach is based on variational methods, a monotonicity trick related to the mountain pass lemma, cut-off functional technique, and a Pohozaev type identity.

We investigate the asymptotic behavior of solutions to the initial boundary value problem for the micropolar fluid model in a half line ℝ+ ≔ (0, ∞). Inspired by the relationship between a micropolar fluid model and Navier-Stokes equations, we prove that the composite wave consisting of the transonic boundary layer solution, the 1-rarefaction wave, the viscous 2-contact wave and the 3-rarefaction wave for the inflow problem on the micropolar fluid model is time-asymptotically stable under some smallness conditions. Meanwhile, we obtain the global existence of solutions based on the basic energy method.

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition$$bq(0,t) + {q_x}(0,t) = 0$$is replaced with a dynamic Robin condition;$$b = b(t)$$is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

This article considers a doubly degenerate parabolic equation with weighted nonlocal sources and nonlinear boundary flux in half-line. We establish the new critical global existence curve and criti...

We obtain new Kolmogorov-type sharp inequalities estimating the norm of the Marchaud fractional derivative $$ {\left\Vert {D}_{-}^kf\right\Vert}_{\infty } $$ of a function f defined on the positive half line in terms of ‖f‖p, 1

Non-self-adjoint second-order differential operators on the half-line with indefinite discontinuous weights are studied. Properties of spectral characteristics are established and inverse problems of recovering operators from their spectral characteristics are investigated. For these class of nonlinear inverse problems algorithms for constructing the global solutions are developed, and uniqueness theorems are proved

The linearization of the classical Boussinesq system is solved explicitly in the case of nonzero boundary conditions on the half-line. The analysis relies on the unified transform method of Fokas and is performed in two different frameworks: (i) by exploiting the recently introduced extension of Fokas's method to systems of equations; (ii) by expressing the linearized classical Boussinesq system as a single, higher-order equation which is then solved via the usual version of the unified transform. The resulting formula provides a novel representation for the solution of the linearized classical Boussinesq system on the half-line. Moreover, thanks to the uniform convergence at the boundary, the novel formula is shown to satisfy the linearized classical Boussinesq system as well as the prescribed initial and boundary data via a direct calculation.

The paper investigates the Urysohn’s nonlinear integral equation on the positive half-line. Some special cases of this equation have specific applications in different areas of modern natural science. In particular, such equations arise in the kinetic theory of gases, in the theory of 𝑝-adic open-closed strings, in mathematical theory of the spatiotemporal spread of the epidemic, and in theory of radiative transfer in spectral lines. The existence theorem for nonnegative nontrivial and bounded solutions is proved. Some qualitative properties of the constructed solution are studied. Specific applied examples of the Urysohn’s kernel satisfying all the conditions of the approved theorem are provided.

Based on the theory of cone and operators, this paper concerned with the existence of unique positive solution for a class of nonlinear fractional boundary value problem with two parameters (one is called an eigenvalue parameter and another is a disturbance parameter) on the half-line. More important, the solutions dependence on two parameters was discussed, which shows that different parameters have different effects on the properties of positive solutions, and the results reflect an interesting fact different from our inference. Some examples are given to illustrate the main results.

An analytical framework for the extension of solutions to fractional differential equations (FDEs) to the negative half-line is presented in this paper. The proposed technique is based on the construction of a special characteristic equation corresponding to the original FDE (when the characteristic equation does exist). This characteristic equation enables the construction analytic solutions to FDEs are expressed in the form of infinite fractional power series. Necessary and sufficient conditions for the existence of such an extension are discussed in detail. It is demonstrated that the extension of solutions to FDEs to the negative half-line is not a single-valued operation. Computational experiments are used to illustrate the efficacy of the proposed scheme.

The authors establish sufficient conditions for the existence of solutions to a boundary value problem for fractional differential inclusions involving the Caputo-Hadamard type derivative of order $ r \in (1, 2] $ on infinite intervals. Both cases of convex and nonconvex valued right hand sides are considered. The technique of proof involves fixed point theorems combined with a diagonalization method.

We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line \begin{document}$ u_t = (u^m)_{xx}+a(x) u^p, $\end{document} $ m, p > 0 $ and $ a(x) = 1 $ for $ x > 0 $, $ a(x) = 0 $ for $ x < 0 $. We first characterize the global existence exponent $ p_0 = 1 $ and the Fujita exponent $ p_c = m+2 $. Then we pass to study the grow-up rate in the case $ p\le1 $ and the blow-up rate for $ p > 1 $. In particular we show that the grow-up rate is different as for global reaction if $ p > m $ or $ p = 1\neq m $.

Gabor systems play a vital role not only in modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this paper, we investigate Gabor systems on positive half line via Walsh-Fourier transform. We provide the complete characterization of orthogonal Gabor systems on positive half line. Furthermore, we provide the characterization of tight frames and orthonormal bases of Gabor systems on positive half line in Fourier domain.

In this paper we study an interacting two-particle system on the positive half-line. We focus on spectral properties of the Hamiltonian for a large class of two-particle potentials. We characterize the essential spectrum and prove, as a main result, the existence of eigenvalues below the bottom of it. We also prove that the discrete spectrum contains only finitely many eigenvalues.

We prove that the law of a random walk $X_n$ is determined by the one-dimensional distributions of $\max(X_n, 0)$ for $n = 1, 2, \ldots$, as conjectured recently by Loic Chaumont and Ron Doney. Equivalently, the law of $X_n$ is determined by its upward space-time Wiener-Hopf factor. Our methods are complex-analytic.