## 1 Introduction
The method suggested in present paper belongs to this group. Differences compared to numerical methods:  
* Simplicity of structure modeling. Equations of equilibrium are solved once independently of the boundary conditions and solution remains unchanged throughout the calculation process. This allows the reduction of structure modeling to set kinematic and static boundary conditions in the separate points at the plate contour. * Possibility to provide better accuracy in comparison with finite element method. It is achieved only by increasing of number of the approximations (parameter \(K\)) or replace nodes at the contour. This procedure is performed automatically with author's program to solve of the plate structure. * Possibility to define the load as a function of two variables. Because not all FEA programs provide such functionality, ABAQUS software have been chosen to compare the results. * Possibility of implementing of suggested method in higher-order theory, say Mindlin-Reissner theory. The mathematical model proposed in the article has been implemented as a computer program which is characterized by a readable and concise source code (Appendix B). This method requires much less computation operations to solve the structure in comparison to FEM. There is no need to:  
* Discretize of the structure into set of finite elements,
* Aggregate of this set into discrete structure,
* Introduce local and global coordinate system,
* Numerate of nodes. Solution of equilibrium equation can be obtained with given accuracy and it is independent of the number of approximations. In contrast to FEM, the suggested method uses essentially fewer equations to solve the structure. Owing to high accuracy of solution of equilibrium equation we need fewer nodes to satisfy boundary conditions. These nodes are placed only at the contour. There are four main techniques to calculate derivatives in the literature [19]: hand-coded analytical derivatives, numerical differentiation, symbolic differentiation and automatic differentiation. Hand-coded analytical derivatives are exact but time consuming to code, error prone and not automated. Numerical differentiation is easy to code. However, as an estimation approach, it is known to produce inaccurate results. Symbolic differentiation is a technique for computing derivatives of mathematical expressions via symbolic manipulation. This technique is used by computer algebra systems. It is exact, but memory intensive and slow. In author's computer program automatic differentiation (AD) is used, because it is free from the drawbacks of techniques described above. Unlike symbolic differentiation, which operates on mathematical expressions, automatic differentiation operates on code. Automatic differentiation leverages the chain-rule-based for evaluating the derivatives. It is exact and speed in comparable to hand-coded derivatives. We calculate higher-order derivatives using _autograd_ library. Finally, using _jax_[20]--the successor of the _autograd_ library--it is possible to run our Python and NumPy programs not only on CPUs (central processing units) but also on much more efficient GPUs (graphics processing units) and TPUs (tensor processing units). Therefore, significant acceleration of calculations can be expected. The purpose of this paper is construction of effective analytical-numerical method to solve of thin symmetrical plates. The suggested method is based on continuum material model. General solution of equation of equilibrium (1) is presented as sum of general solution of uniform Equation (2) and particular solution of non-uniform Equation (1). They are independent between themselves and contain theoretically infinite number of unknown parameters which allow the performance of the equation of equilibrium and boundary conditions with given accuracy. General solution of equilibrium equation includes functions of shape of deflection. Functions of shape of displacements, moments, shearing forces and generalized shearing forces have been obtained using automatic differentiation.