(R1885) Analytical and Numerical Solutions of a Fractional-Order Mathematical Model of Tumor Growth for Variable Killing Rate

This work intends to analyze the dynamics of the most aggressive form of brain tumor, glioblastomas, by following a fractional calculus approach. In describing memory preserving models, the non-local fractional derivatives not only deliver enhanced results but also acknowledge new avenues to be further explored. We suggest a mathematical model of fractional-order Burgess equation for new research perspectives of gliomas, which shall be interesting for biomedical and mathematical researchers. We replace the classical derivative with a non-integer derivative and attempt to retrieve the classical solution as a particular case. The prime motive is to acquire both analytical and numerical solutions to the posed problem. At first, we employ the transform method, and then the Adomian decomposition method to obtain the solutions that shall be useful to provide information about the effect of medical care in the annihilation of gliomas. Finally, we discuss the applicability of this model with numerical simulations and graphical representations

Computational and numerical analysis of differential equations using spectral based collocation method.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.In this thesis, we develop accurate and computationally efficient spectral collocation-based methods, both modified and new, and apply them to solve differential equations. Spectral collocation-based methods are the most commonly used methods for approximating smooth solutions of differential equations defined over simple geometries. Procedurally, these methods entail transforming the gov erning differential equation(s) into a system of linear algebraic equations that can be solved directly. Owing to the complexity of expanding the numerical algorithms to higher dimensions, as reported in the literature, researchers often transform their models to reduce the number of variables or narrow them down to problems with fewer dimensions. Such a process is accomplished by making a series of assumptions that limit the scope of the study. To address this deficiency, the present study explores the development of numerical algorithms for solving ordinary and partial differential equations defined over simple geometries. The solutions of the differential equations considered are approximated using interpolating polynomials that satisfy the given differential equation at se lected distinct collocation points preferably the Chebyshev-Gauss-Lobatto points. The size of the computational domain is particularly emphasized as it plays a key role in determining the number of grid points that are used; a feature that dictates the accuracy and the computational expense of the spectral method. To solve differential equations defined on large computational domains much effort is devoted to the development and application of new multidomain approaches, based on decomposing large spatial domain(s) into a sequence of overlapping subintervals and a large time interval into equal non-overlapping subintervals. The rigorous analysis of the numerical results con firms the superiority of these multiple domain techniques in terms of accuracy and computational efficiency over the single domain approach when applied to problems defined over large domains. The structure of the thesis indicates a smooth sequence of constructing spectral collocation method algorithms for problems across different dimensions. The process of switching between dimensions is explained by presenting the work in chronological order from a simple one-dimensional problem to more complex higher-dimensional problems. The preliminary chapter explores solutions of or dinary differential equations. Subsequent chapters then build on solutions to partial differential i equations in order of increasing computational complexity. The transition between intermediate dimensions is demonstrated and reinforced while highlighting the computational complexities in volved. Discussions of the numerical methods terminate with development and application of a new method namely; the trivariate spectral collocation method for solving two-dimensional initial boundary value problems. Finally, the new error bound theorems on polynomial interpolation are presented with rigorous proofs in each chapter to benchmark the adoption of the different numerical algorithms. The numerical results of the study confirm that incorporating domain decomposition techniques in spectral collocation methods work effectively for all dimensions, as we report highly accurate results obtained in a computationally efficient manner for problems defined on large do mains. The findings of this study thus lay a solid foundation to overcome major challenges that numerical analysts might encounter

Integration of the hyperbolic telegraph equation in (1+1) dimensions via the generalized differential quadrature method

AbstractThe 2D generalized differential quadrature method (hereafter called ((1+1)-GDQ) is introduced within the context of dynamical system for solving the hyperbolic telegraph equation in (1+1) dimensions. Best efficiency is obtained with a low-degree polynomial (n⩽8) for both time variable and x-direction. From realistic examples, some models are presented to illustrate an excellent performance of the proposed method, compared with the exact results

High-Order Numerical Solution of Second-Order One-Dimensional Hyperbolic Telegraph Equation Using a Shifted Gegenbauer Pseudospectral Method

We present a high-order shifted Gegenbauer pseudospectral method (SGPM) to solve numerically the second-order one-dimensional hyperbolic telegraph equation provided with some initial and Dirichlet boundary conditions. The framework of the numerical scheme involves the recast of the problem into its integral formulation followed by its discretization into a system of well-conditioned linear algebraic equations. The integral operators are numerically approximated using some novel shifted Gegenbauer operational matrices of integration. We derive the error formula of the associated numerical quadratures. We also present a method to optimize the constructed operational matrix of integration by minimizing the associated quadrature error in some optimality sense. We study the error bounds and convergence of the optimal shifted Gegenbauer operational matrix of integration. Moreover, we construct the relation between the operational matrices of integration of the shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive the global collocation matrix of the SGPM, and construct an efficient computational algorithm for the solution of the collocation equations. We present a study on the computational cost of the developed computational algorithm, and a rigorous convergence and error analysis of the introduced method. Four numerical test examples have been carried out in order to verify the effectiveness, the accuracy, and the exponential convergence of the method. The SGPM is a robust technique, which can be extended to solve a wide range of problems arising in numerous applications.Comment: 36 pages, articl

Incorporating Inductances in Tissue-Scale Models of Cardiac Electrophysiology

In standard models of cardiac electrophysiology, including the bidomain and monodomain models, local perturbations can propagate at infinite speed. We address this unrealistic property by developing a hyperbolic bidomain model that is based on a generalization of Ohm's law with a Cattaneo-type model for the fluxes. Further, we obtain a hyperbolic monodomain model in the case that the intracellular and extracellular conductivity tensors have the same anisotropy ratio. In one spatial dimension, the hyperbolic monodomain model is equivalent to a cable model that includes axial inductances, and the relaxation times of the Cattaneo fluxes are strictly related to these inductances. A purely linear analysis shows that the inductances are negligible, but models of cardiac electrophysiology are highly nonlinear, and linear predictions may not capture the fully nonlinear dynamics. In fact, contrary to the linear analysis, we show that for simple nonlinear ionic models, an increase in conduction velocity is obtained for small and moderate values of the relaxation time. A similar behavior is also demonstrated with biophysically detailed ionic models. Using the Fenton-Karma model along with a low-order finite element spatial discretization, we numerically analyze differences between the standard monodomain model and the hyperbolic monodomain model. In a simple benchmark test, we show that the propagation of the action potential is strongly influenced by the alignment of the fibers with respect to the mesh in both the parabolic and hyperbolic models when using relatively coarse spatial discretizations. Accurate predictions of the conduction velocity require computational mesh spacings on the order of a single cardiac cell. We also compare the two formulations in the case of spiral break up and atrial fibrillation in an anatomically detailed model of the left atrium, and [...].Comment: 20 pages, 12 figure