Abstract
Abstract With the development of modern partial differential equation (PDE) theory, the theory of linear PDE is becoming more and more perfect, . Non-linear PDE has become a research hotspot of many mathematicians. In fact, when describing practical physical problems with PDEs, non-linear problems tend to be more general than linear problems, which are close to real problems and have practical physical significance. Hyperbolic PDEs are a kind of important PDEs describing the phenomena of vibration or wave motion. The solution of hyperbolic PDE can be decomposed into the form of multiplication of vibration and vibration or of exponential function and exponential function. Generally, the energy is infinite. A full discrete convergence analysis method for non-linear hyperbolic equation based on finite element analysis is proposed. Taking second-order and fourth-order non-linear hyperbolic equation as examples, the full discrete convergence of non-linear hyperbolic equation is analysed by finite element method and the super-convergence results are obtained.
Highlights
With the rapid development of science and technology, a variety of differential equation mathematical models have been pouring out [1,2]
Hydrodynamic problems in aviation, meteorology, ocean, petroleum exploration and other fields are reduced to solving non-linear hyperbolic partial differential equations (PDEs; known as conservation laws in foreign literature)
In the process of neural propagation, neural transmission signals and the rate of change in time and space are mathematically represented as a class of initial boundary value problems for non-linear quasi-hyperbolic equations
Summary
With the rapid development of science and technology, a variety of differential equation mathematical models have been pouring out [1,2]. The hyperbolic equation (group) model is one of the most important ones It has a wide application background in natural science. It belongs to one-dimensional wave equation describing string vibration. The Maxwell equations describing electromagnetic fields are curled to simplify the standard vector wave equations [5]. Hydrodynamic problems in aviation, meteorology, ocean, petroleum exploration and other fields are reduced to solving non-linear hyperbolic partial differential equations (PDEs; known as conservation laws in foreign literature). Hyperbolic equations (systems) are widely used in many fields of mathematical physics and have profound physical background, such as wave equation. They have been paid more attention by mathematicians and engineering technicians. The full discrete convergence of the non-linear hyperbolic equation is analysed comprehensively [10]
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