12 - Numerical method for a fractal–fractional ordinary differential equation with variable order with power-law kernel

Abstract

Power has played a very important role in modeling real world problems, especially those following power law processes. However, many research used power law with constant fractional orders. On the other hand, the concept of fractal differentiation has been introduced and have been found suitable mathematical tool to modeling some complex problems. However, the fractal dimension or orders are constant. Constant variable orders derivative have been found inadequate to modeling anomalous diffusions. In this chapter, we consider differential equations with fractal–fractional variable order fractal dimension and constant fractional order. In this chapter, two step-Newton polynomial interpolation is used to generate numerical scheme that could be used to solve numerically these classes of ordinary differential equations.