Non-differentiable kernel-based approximation of memory-dependent derivative for drug delivery applications

Abstract

Abstract Although memory-dependent derivative (MDD) has recently been the subject of extensive study, only one numerical approximation has been reported in the literature. Hence, this study introduces a novel approximation for MDD. Moreover, a new form of the kernel function is presented. The convergence order of our approximation is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">O</mml:mi> <mml:mfenced close=")" open="("> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>h</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>−</mml:mo> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">S</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">S</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">S</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> is the exponent of the kernel function. The proposed approach is used to numerically solve the memory-dependent advection-diffusion problem, and the numerical scheme's stability and convergence are discussed. Also, memory-based models have been described to study drug delivery and its diffusion from multi-layer capsules/tablets. These models are based on the fractional derivative and the MDD to solve the paradox of the unphysical feature of the infinite propagation speed of the published Fickian-based models. The theoretical analysis is validated with numerical examples by investigating the convergence order that is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>−</mml:mo> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">S</mml:mi> </mml:mrow> </mml:mfenced> </mml:math> in time. To illustrate the validity and efficiency of the proposed models, profiles of concentration and drug mass are compared with the observed in vivo data. It is observed to be highly accurate and compatible with the in vivo data when compared with Fick's model.