*Correspondence: [email protected] Institute of Applied Physics and Computational Mathematics, Beijing, 100088, P.R. China College of Mathematics, Jilin University, Changchun, 130012, P.R. China Abstract In this paper, we establish the Nagumo theorems for boundary value problems associated with a class of third-order singular nonlinear equations: (p(t)x′)′′ = f (t, x,p(t)x′, (p(t)x′)′), ∀t ...

In this paper, we present the Green’s functions for a second-order linear differential equation with three-point boundary conditions. We give exact expressions of the solutions for the linear three-point boundary problems by the Green’s functions. As applications, we study uniqueness and iteration of the solutions for a nonlinear singular second-order three-point boundary value problem.

This paper discusses the existence of solutions of the fractional differential equations D(φ(Du)) = Fu, D(φ(Du)) = f(t, u, Du) satisfying the boundary conditions u(0) = A(u), u(T ) = B(u). Here μ, α ∈ (0, 1], ν ∈ (0, α], D is the Caputo fractional derivative, φ ∈ C(−a, a) (a > 0), F is a continuous operator, A,B are bounded and continuous functionals and f ∈ C([0, T ] × R). The existence result...

In this report, we discuss the implementation and numerical aspects of the Matlab solver sbvp designed for the solution of two-point boundary value problems, which may include a singularity of the first kind, z′(t) = f(t, z(t)) := 1 (t− a) · z(t) + g(t, z(t)) , t ∈ (a, b), R(z(a), z(b)) = 0. The code is based on collocation at either equidistant or Gaussian collocation points. For singular prob...

where f(t),g(t), and h(t) are known continuous functions of t in the interval (0,1). Here N(u) is a nonlinear function of u. Let the above equation be singular at these two boundary value points t= 0,1. Scientists and engineers are interested in singular BVPs because they arise in a wide range of applications, such as in chemical engineering, mechanical engineering, nuclear industry, and nonlin...

For each n ∈ N, n ≥ 2 we prove the existence of a positive solution of the singular discrete problem 1 h2 ∆uk−1 + f(tk, uk) = 0, k = 1, . . . , n− 1, ∆u0 = 0, un = 0, where T ∈ (0,∞), h = T n , tk = hk, f : [0, T ] × (0,∞) is continuous and has a singularity at x = 0. We prove that for n → ∞ the sequence of solutions of the above discrete problems converges to a solution y of the corresponding ...