In this paper, we consider a collocation method for solving a nonlinear Volterra integral equation involving the solution-dependent singular kernel (t−s)−α(u(s))F(s,u(s)), where F(s,u(s))=sσϕ(s,u(s)), 0<σ<1, and ϕ is a suitably defined nonlinear function. We first establish the existence and uniqueness of a continuous solution to the equation using Schaefer’s fixed-point theorem. We then develop a collocation method on a graded mesh for its numerical approximation and derive corresponding error estimates. Finally, numerical experiments are presented to verify the theoretical convergence rates.
We consider a Volterra integral equation with multiscale and nonlinear exponent kernel. We propose the high-order smoothing conditions, under which the initial singularity of the solutions can be eliminated up to any prescribed order. This indicates the application of the multiscale nature of the kernel on local modifi...
Wen-Lin Qiu, Chuan-Wei Su, Xiang-Cheng Zheng· 0 citations
In this paper, we study the existence, uniqueness and homogenization of a nonlinear hyperbolic problem with nonlinear boundary damping term [Formula: see text] and a boundary source term [Formula: see text] where [Formula: see text] At first, we prove the existence and uniqueness of a solution related to this problem t...
Dehamnia Abdelhakim, Haddadou Hamid· Journal of Hyperbolic Differ...· 0 citations
We study a nonlinear Dirichlet boundary value problem with the variable exponent p(x)-Laplacian and a reaction term. First, we prove the existence and uniqueness of a weak solution by minimizing the associated energy functional. To deal with possible singularities in the operator, we introduce a relaxed energy through...
We study the fourth-order elliptic equations of Kirchhoff-type Δ2u−(a+b∫ℝN|∇u|2dx)Δu+V(x)u=f(x,u)+λh(x,u),x∈ℝN, where a>0, b≥0 are constants, we have the potential V(x):ℝN→ℝ, V∈C(ℝN,ℝ). The nonlinearity λh(x,u)+f(x,u) may involve a combination of concave and convex terms. Under some suitable conditions on h,f∈C(ℝN×ℝ,ℝ)...
The aim of this paper is to investigate the existence, uniqueness, and Hyers–Ulam stability of solutions to a class of nonlinear Caputo fractional delay differential equation with nonlocal boundary conditions in the space of absolutely continuous functions. Owing to the nonlinearity, the equation is not readily amenabl...
We investigate the application of collocation schemes to a class of Volterra delay integral equations that feature compact and noncompact operators. This class of integral equations with vanishing delays is solved numerically using multi-step collocation techniques implemented on various mesh types. The study includes...
Y. J. Subhi, P. Darania, S. Pishbin· Malaysian journal of mathema...· 0 citations
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