Investigation of Fractional Order Tumor Cell Concentration Equation Using Finite Difference Method

Abstract

  The fractional differential equation provides a powerful mathematical framework for modeling and understanding a wide range of complex, non-local and memory-dependent phenomena in various scientific, engineering and real-world applications. Liver metastasis is a secondary cancerous tumor that develops in the liver due to the spread of cancer cells from a primary cancer originating in another part of the human body. The primary focus of this study is to understand tumor growth in the human liver, both in the presence and absence of medication therapy. To achieve this, a temporal fractional-order parabolic partial differential equation is utilized, and its analysis is carried out using numerical methods. The Caputo derivative is employed to explore the impact of medication therapy on tumor growth. To numerically solve the mathematical model, the Crank-Nicolson finite difference method has been developed. This method is selected for its remarkable attributes, including unconditional stability and second-order accuracy in both spatial and temporal dimensions. The outcomes and insights derived from this study are effectively communicated through various graphical representations. These visual aids serve as invaluable tools for comprehending the profound impact of medication therapy on the growth of liver tumors. Through the medium of these graphical depictions, one can glean a clearer and more intuitive understanding of the complex dynamics at play. The numeric solution to this intricate problem is achieved through the implementation of algorithms meticulously crafted using versatile and powerful Python programming language. Python’s flexibility, extensive libraries, and robust numerical computing capabilities make it a good choice for handling the complexities of this study.