Obtaining Approximate Solutions of Differential Equations Using the Method of Moving Nodes

The article discusses obtaining an approximate analytical solution to the Cauchy problem for an ordinary differential equation and an initial-boundary value problem for a linear parabolic equation. Obtaining an approximate analytical solution is based on the method of moving nodes. Various numerical methods are known for such problems. Using explicit and implicit Euler methods for solving the Cauchy problem in combination with ideas from the method of moving nodes, the possibility of obtaining an approximate analytical form of solving the problem is indicated. To refine the solution, a multi-point moving node was used. The method of using a multi-point movable unit taking into account linearization allows obtaining an approximate analytical solution to the Cauchy problem. Obtaining an approximate analytical expression for the initial boundary value problem for a one-dimensional parabolic equation is considered in various aspects. For a numerical solution, there are various numerical methods (finite difference method, control volume method, etc.). The finite-difference method, taking into account the movability of the node, allowed us to obtain a simple approximate solution. Approximation of derivatives in a differential equation is carried out in various ways. In the first version, both partial derivatives were approximated by a finite difference with the moving node. Using the resulting equation, an approximate solution was determined. In the second and third options, the approximation was carried out using only one of the variables, and by solving this, we obtained an approximate solution. Various examples are considered and the resulting solutions are compared.

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