Modified Swanson’s equation to detect the growth of glioblastomas multiforme (GBM) tumour

The concentration of glioblastomas multiforme (GBM) tumour equation is numerically solved in terms of net rates of proliferation and invasion. The GBM tumour cells concentration evolution is known as a reaction-diffusion process and the diffusion coefficients differ according to the brain’s white and grey matter composition. An Integer order non-linear equation can produce significant errors as most of the method used to solve the equation require linearization, discretization and perturbation which involved more computational work. Hence, a non-linear diffusion logistic density model proposed by Özuğurlu (2015) is modified to non-linear fractional-order partial differential equation (FPDE) that described in the Caputo sense. This study presents another alternative to investigate the GBM tumour growth using the theory of fractional calculus (FC). Q-homotopy analysis transform method (q-HATM) and Laplace Adomian decomposition method (LADM) were applied to solve the proposed model. From the results obtained, the fractional order (α) has greater flexibility which allows more degree of freedom in design and analysis. Results derived are in more accurate manner. Q-HATM is more efficient as it proposes a modest way to adjust the stability and the convergence region of the solution using an auxiliary parameter (h) and asymptotic parameter (n≥1).

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