Cauchy problem on non-solvable systems of first-order partial differential equations with applications

Let $L_1, L_2, \cdots , L_m$ be $m$ partial differential operators of first order and $h_1, h_2, \cdots , h_m$ continuously differentiable functions. Then is the partial differential equation system $L_i[u] = h_i, 1 \leq i \leq m$ solvable for a differential mapping $u : \mathbb{R}^n \to \mathbb{R}^n$ or not? Similarly, let $\varphi$ be a continuous function $\varphi : \mathbb{R}^{n-1} \to \mathbb{R}^{n-1}$. Then is the Cauchy problem $L_i[u] = h_i, 1 \leq i \leq m$ with $u(x_1, x_2, \cdots , x_{n-1}, t_0) = \varphi (x_1, x_2, \cdots , x_{n-1})$ solvable or not? If not, how can we characterize the behavior of such a function $u$? All these questions are ignored in classical mathematics only by saying not solvable! In fact, non-solvable equation systems are nothing but Smarandache systems, i.e., contradictory systems themselves, in which a ruler behaves in at least two different ways within the same system, i.e., validated and invalided, or only invalided but in multiple distinct ways. They are widely existing in the natural world and our daily life. In this paper, we discuss non-solvable partial differential equation systems of first order by a combinatorial approach, classify these systems by underlying graphs, particularly, these non-solvable linear systems, characterize their behaviors, such as those of global stability, energy integral and their geometry, which enables one to find a differentiable manifold with preset $m$ vector fields. Applications of such non-solvable systems to interaction fields and flows in network are also included in this paper.

Keywords

non-solvable partial-differential equation, vertex-edge labeled graph, global stability, energy integral, combinatorial manifold

2010 Mathematics Subject Classification

05C15, 34A30, 34A34, 37C75, 70F10, 92B05

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Published 1 June 2015