Communications in Nonlinear Science and Numerical Simulation, 2009
ABSTRACT In this paper, we develop quintic nonpolynomial spline methods for the numerical solutio... more ABSTRACT In this paper, we develop quintic nonpolynomial spline methods for the numerical solution of fourth order two-point boundary value problems. Using this spline function a few consistency relations are derived for computing approximations to the solution of the problem. The present approach gives better approximations and generalizes all the existing polynomial spline methods up to order four. This approach has less computational cost. Convergence analysis of these methods is discussed. Two numerical examples are included to illustrate the practical usefulness of our methods.
International Journal for Numerical Methods in Biomedical Engineering, Jul 10, 2023
Mathematical simulation of drug diffusion is a significant tool for predicting the bio‐transport ... more Mathematical simulation of drug diffusion is a significant tool for predicting the bio‐transport process. Moreover, the reported models in the literature are based on Fick's approach, which leads to an infinite propagation speed. Consequently, it is essential to construct a mathematical model to represent the diffusion processes for estimating drug concentrations at different sites and throughout the circulation. Thus, in this article, the diffusion process is employed to propose three models for estimating the drug release from multi‐layer cylindrical tablets. A fractional model is presented based on Fick's approach, while classical and fractional Cattaneo models are presented using the relaxed principle. Various numerical methods are used to solve the specified problem. The numerical scheme's stability and convergence are demonstrated. Drug concentration and mass profiles are presented for the tablet and the external medium and compared with the in vivo plasma profiles. The results show the efficiency and precision of the proposed fractional models based on the fourth‐order weighted‐shifted Grünwald–Letnikov difference operator approximation. These models are compatible with the in vivo data compared with the classical Fick's one.
Communications in Nonlinear Science and Numerical Simulation, 2009
ABSTRACT In this paper, we develop quintic nonpolynomial spline methods for the numerical solutio... more ABSTRACT In this paper, we develop quintic nonpolynomial spline methods for the numerical solution of fourth order two-point boundary value problems. Using this spline function a few consistency relations are derived for computing approximations to the solution of the problem. The present approach gives better approximations and generalizes all the existing polynomial spline methods up to order four. This approach has less computational cost. Convergence analysis of these methods is discussed. Two numerical examples are included to illustrate the practical usefulness of our methods.
International Journal for Numerical Methods in Biomedical Engineering, Jul 10, 2023
Mathematical simulation of drug diffusion is a significant tool for predicting the bio‐transport ... more Mathematical simulation of drug diffusion is a significant tool for predicting the bio‐transport process. Moreover, the reported models in the literature are based on Fick's approach, which leads to an infinite propagation speed. Consequently, it is essential to construct a mathematical model to represent the diffusion processes for estimating drug concentrations at different sites and throughout the circulation. Thus, in this article, the diffusion process is employed to propose three models for estimating the drug release from multi‐layer cylindrical tablets. A fractional model is presented based on Fick's approach, while classical and fractional Cattaneo models are presented using the relaxed principle. Various numerical methods are used to solve the specified problem. The numerical scheme's stability and convergence are demonstrated. Drug concentration and mass profiles are presented for the tablet and the external medium and compared with the in vivo plasma profiles. The results show the efficiency and precision of the proposed fractional models based on the fourth‐order weighted‐shifted Grünwald–Letnikov difference operator approximation. These models are compatible with the in vivo data compared with the classical Fick's one.
B-spline functions introduce a great importance in several science
branches as in numerical analy... more B-spline functions introduce a great importance in several science branches as in numerical analysis, ordinary and partial differential equations, integral equations and statistical analysis. It has also many applications in science, engineering, economics, biology and medicine, etc. In this book, we study cubic B-spline in calculating numerical solutions for second order parabolic partial differential equations. In addition, we use quartic B-spline in calculating numerical solutions for third order nonlinear partial differential equations. Moreover, sextic B-splines are used to solve fifth order partial differential equations.
Uploads
Papers by waheed K zahra
branches as in numerical analysis, ordinary and partial differential equations,
integral equations and statistical analysis. It has also many applications in
science, engineering, economics, biology and medicine, etc.
In this book, we study cubic B-spline in calculating numerical
solutions for second order parabolic partial differential equations. In addition,
we use quartic B-spline in calculating numerical solutions for third order
nonlinear partial differential equations. Moreover, sextic B-splines are used
to solve fifth order partial differential equations.