Class Of Nonlinear Fourth-order Equations Research Articles

An Implicit Difference Scheme for the Fourth-Order Nonlinear Evolution Equation with Multi-Term Riemann–Liouvile Fractional Integral Kernels

In this paper, an implicit difference scheme is proposed and analyzed for a class of nonlinear fourth-order equations with the multi-term Riemann–Liouvile (R–L) fractional integral kernels. For the nonlinear convection term, we handle implicitly and attain a system of nonlinear algebraic equations by using the Galerkin method based on piecewise linear test functions. The Riemann–Liouvile fractional integral terms are treated by convolution quadrature. In order to obtain a fully discrete method, the standard central difference approximation is used to discretize the spatial derivative. The stability and convergence are rigorously proved by the discrete energy method. In addition, the existence and uniqueness of numerical solutions for nonlinear systems are proved strictly. Additionally, we introduce and compare the Besse relaxation algorithm, the Newton iterative method, and the linearized iterative algorithm for solving the nonlinear systems. Numerical results confirm the theoretical analysis and show the effectiveness of the method.

Existence of bounded solutions for a class of nonlinear fourth-order equations

In this article, we consider a class of nonlinear elliptic fourth-order equations with the principal part satisfying a strengthened coercivity condition. It is supposed that the lower- order term of the equations admits an arbitrary growth with respect to unknown function and the growth rates of derivatives of this function coinciding with the exponents of the corresponding energy space. We prove a theorem on existence of bounded generalized solutions of the Dirichlet problem for equations of the given class.