In this paper, we propose the representation and cohomology of modified $\lambda$-differential 3-Lie algebras. As their applications, the linear deformations, abelian extensions and $T^{\ast }$-extensions of modified $\lambda$-differential 3-Lie algebras are also studied.

Under simply supported plate (SSP) boundary conditions, a numerical method based on the higher-order Legendre polynomial approximation was studied and developed for fourth-order problems with variable coefficients. We first divide the SSP boundary conditions into two types, namely, forced boundary conditions and natural boundary conditions. According to the forced boundary conditions, an appropriate Sobolev space is defined, and a variational formulation and a discrete scheme associated with the original problem are established. Then, the existence and uniqueness of this weak solution and approximate solution are proved. By using the Céa lemma and the tensor Jacobian polynomial approximation, we further obtain the error estimation for the numerical solutions. In addition, we use the orthogonality of Legendre polynomials to construct a set of effective basis functions and derive the equivalent tensor product linear system associated with the discrete scheme, respectively. Finally, some numerical tests were carried out to validate our algorithm and theoretical analysis.