### Boundary value problems for the mildly non-linear ordinary differential equation of the fourth order

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The present paper deals with numerical solution of a viscoelastic plate. The discrete problem is defined by ${C}^{1}$-elements and a linear multistep method. The effect of numerical integration is studied as well. The rate of cnvergence is established. Some examples are given in the conclusion.

We describe a numerical method for the equation ${u}_{t}+p{u}_{x}-\epsilon {u}_{xx}=f$ in $(0,1)\times (0,T)$ with Dirichlet boundary and initial conditions which is a combination of the method of characteristics and the finite-difference method. We prove both an a priori local error-estimate of a high order and stability. Example 3.3 indicates that our approximate solutions are disturbed only by a minimal amount of the artificial diffusion.

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