On a nonlocal boundary value problem for the equation fourth-order in partial derivatives

В данной статье изучается нелокальная задача для уравнения четвертого порядка в которой доказывается существование и единственность решения этой задачи. Решение построено явно в виде ряда Фурье, обоснованы абсолютная и равномерная сходимость полученного ряда и возможность почленного дифференцирования решения по всем переменным. Установлен критерий однозначной разрешимости поставленной краевой задачи. In this article, we study a nonlocal problem for a fourth-order equation in which the existence and uniqueness of a solution to this problem is proved. The solution is constructed explicitly in the form of a Fourier series; the absolute and uniform convergence of the obtained series and the possibility of term-by-term differentiation of the solution with respect to all variables are substantiated. A criterion for the unique solvability of the stated boundary value problem is established.

Existence of Positive Solutions for a Pertubed Fourth-Order Equation

In this paper, a special type of fourth-order differential equations with a perturbed nonlinear term and some boundary conditions is considered which is very important in mechanical engineering. Therefore, the existence of a non-trivial solution for such equations is very important. Our goal is to ensure at least three weak solutions for a class of perturbed fourth-order problems by applying certain conditions to the functions that are available in the differential equation (problem (??)). Our approach is based on variational methods and critical point theory. In fact, using a fundamental theorem that is attributed to Bonanno, we get some important results. Finally, for some results, an example is presented.

The Galerkin Method for Fourth-Order Equation of the Moore–Gibson–Thompson Type with Integral Condition

In this manuscript, we consider the fourth order of the Moore–Gibson–Thompson equation by using Galerkin’s method to prove the solvability of the given nonlocal problem.

Sign-changing solutions for Schrödinger–Kirchhoff-type fourth-order equation with potential vanishing at infinity

The purpose of this paper is to study the existence of sign-changing solution to the following fourth-order equation: $$ \Delta ^{2}u- \biggl(a+ b \int _{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \Delta u+V(x)u=K(x)f(u) \quad\text{in } \mathbb{R}^{N}, $$ Δ 2 u − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + V ( x ) u = K ( x ) f ( u ) in R N , where $5\leq N\leq 7$ 5 ≤ N ≤ 7 , $\Delta ^{2}$ Δ 2 denotes the biharmonic operator, $K(x), V(x)$ K ( x ) , V ( x ) are positive continuous functions which vanish at infinity, and $f(u)$ f ( u ) is only a continuous function. We prove that the equation has a least energy sign-changing solution by the minimization argument on the sign-changing Nehari manifold. If, additionally, f is an odd function, we obtain that equation has infinitely many nontrivial solutions.

Well-posedness for a fourth-order equation of Moore–Gibson–Thompson type

In this paper, we completely characterize, only in terms of the data, the well-posedness of a fourth order abstract evolution equation arising from the Moore–Gibson–Thomson equation with memory. This characterization is obtained in the scales of vector-valued Lebesgue, Besov and Triebel–Lizorkin function spaces. Our characterization is flexible enough to admite as examples the Laplacian and the fractional Laplacian operators, among others. We also provide a practical and general criteria that allows Lp–Lq-well posedness.

Analysis of optimal superconvergence of an ultraweak-local discontinuous Galerkin method for a time dependent fourth-order equation

In this paper, we study superconvergence properties of the ultraweak-local discontinuous Galerkin (UWLDG) method in Tao et al. [To appear in Math. Comput. DOI: https://doi.org/10.1090/mcom/3562 (2020).] for an one-dimensional linear fourth-order equation. With special initial discretizations, we prove the numerical solution of the semi-discrete UWLDG scheme superconverges to a special projection of the exact solution. The order of this superconvergence is proved to be k + min(3, k) when piecewise ℙk polynomials with k ≥ 2 are used. We also prove a 2k-th order superconvergence rate for the cell averages and for the function values and derivatives of the UWLDG approximation at cell boundaries. Moreover, we prove superconvergence of (k + 2)-th and (k + 1)-th order of the function values and the first order derivatives of the UWLDG solution at a class of special quadrature points, respectively. Our proof is valid for arbitrary non-uniform regular meshes and for arbitrary k ≥ 2. Numerical experiments verify that all theoretical findings are sharp.