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The main goal of this paper is to present a novel numerical scheme based on the Fibonacci wavelets for solving the brain tumor growth model governed by the Burgess equation. At the first instance, the Fibonacci-wavelet-based operational matrices of integration are obtained by following the well-known Chen–Hsiao technique. These matrices play a vital role in converting the said model into an algebraic system, which could be handled with any standard numerical method. To access the effect of medical treatment over the brain tumor growth, we have investigated both the linear and nonlinear cases of Burgess equation. The nonlinearity arising in the Burgess equation is handled by invoking the quasilinearization technique. In order to compare the efficiency of the Fibonacci-wavelet-based numerical technique, we formulated an analogous numerical scheme based on the Haar wavelets. Subsequently, both the methods are testified on several test problems and it is demonstrated that the Fibonacci wavelet method yields a much more stable solution and a better approximation than the Haar wavelet method.

In this paper, we develop an efficient wavelet method based on the Fibonacci polynomials for solving the Pennes bioheat transfer equation. The formulation of the proposed technique is started with the construction of Fibonacci wavelets by using Fibonacci polynomials and then applying spectral collocation technique to transform the given problem into a system of an algebraic equation, that can be solved using the Newton method. Some results related to error estimate and convergence analysis of the proposed scheme are also investigated. The applicability and accuracy of the present technique are elucidated by a comparison with the exact solution and those of the other methods found in the recent literature. The obtained results show that the proposed technique is an effective tool for solving Pennes bioheat transfer equations and can also be used for solving similar types of various partial differential equations numerically.