This paper proposes an explicit Fourier-Klibanov method as a new approximation technique for an age-dependent population PDE of Gompertz type in modeling the evolution of tumor density in a brain tissue. Through suitable nonlinear and linear transformations, the Gompertz model of interest is transformed into an auxiliary third-order nonlinear PDE. Then, a coupled transport-like PDE system is obtained via an application of the Fourier-Klibanov method, and, thereby, is approximated by the explicit finite difference operators of characteristics. The stability of the resulting difference scheme is analyzed under the standard 2-norm topology. Finally, we present some computational results to demonstrate the effectiveness of the proposed method.

翻译：本文提出了一种显式Fourier-Klibanov方法,作为Gompertz型年龄依赖种群偏微分方程的新型逼近技术,用于模拟脑组织中肿瘤密度的演化过程。通过适当的非线性和线性变换,所关注的Gompertz模型被转化为辅助的三阶非线性偏微分方程。随后,应用Fourier-Klibanov方法得到耦合的输运型偏微分方程组,并利用特征方向的显式有限差分算子对其进行逼近。在标准2-范数拓扑下分析了所得差分格式的稳定性。最后,我们给出了若干计算结果表明所提方法的有效性。