Abstract It is well known that the simulation of fractional systems is a difficult task from all points of view. In particular, the computer implementation of numerical algorithms to simulate fractional systems of partial differential equations in three dimensions is a hard task which has no been solved satisfactorily. In this work, we propose a numerical method to solve systems of hyperbolic (fractional o non-fractional) partial differential equations that generalize various known models from physics, chemistry and biology. The scheme is an explicit technique which has the advantage of being easy to implement for any scientist with minimal knowledge on scientific programming. We propose a computer implementation which exploits the advantages of the efficient matrix algebra already available in Fortran and other languages. The algorithm is presented mathematically as well as in pseudo-code, and a raw Fortran implementation of the computer algorithm is provided in the appendix. This code is susceptible to be compiled in parallel using OpenMP, whence it follows that the computer time can be substantially reduced. As application, we provide some illustrative simulations on the formation of Turing patterns in a three-dimensional system of inhibitor–activator substances in physics. The graphs were obtained using functions of Matlab with the numerical outputs generated by our Fortran code.

中文翻译：

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一种易于实现的并行算法，用于模拟三维（分数）双曲系统中的复杂不稳定性

摘要 众所周知，分数系统的仿真从各个角度来看都是一项艰巨的任务。尤其是计算机实现数值算法来模拟三维偏微分方程的分数系统是一项艰巨的任务，尚未得到令人满意的解决。在这项工作中，我们提出了一种数值方法来求解双曲（分数或非分数）偏微分方程系统，这些方程概括了物理学、化学和生物学中的各种已知模型。该方案是一种显式技术，其优点是对于任何对科学编程知识知之甚少的科学家来说都很容易实现。我们提出了一种计算机实现，它利用了 Fortran 和其他语言中已有的高效矩阵代数的优点。该算法以数学和伪代码形式呈现，附录中提供了计算机算法的原始 Fortran 实现。此代码易于使用 OpenMP 并行编译，因此可以显着减少计算机时间。作为应用，我们提供了一些关于物理学中抑制剂-活化剂物质三维系统中图灵模式形成的说明性模拟。这些图形是使用 Matlab 的函数和由我们的 Fortran 代码生成的数字输出获得的。我们提供了一些关于物理学中抑制剂-活化剂物质三维系统中图灵模式形成的说明性模拟。这些图形是使用 Matlab 的函数和由我们的 Fortran 代码生成的数字输出获得的。我们提供了一些关于物理学中抑制剂-活化剂物质三维系统中图灵模式形成的说明性模拟。这些图形是使用 Matlab 的函数和由我们的 Fortran 代码生成的数字输出获得的。