Conformable Derivative Operator in Modelling Neuronal Dynamics

This study presents two new numerical techniques for solving time-fractional one-dimensional cable differential equation (FCE) modeling neuronal dynamics. We have introduced new formulations for the approximate-analytical solution of the FCE by using modified homotopy perturbation method defined with conformable operator (MHPMC) and reduced differential transform method defined with conformable operator (RDTMC), which are derived the solutions for linear-nonlinear fractional PDEs. In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of fractional neuronal dynamics problem. Moreover, we have declared that the proposed models are very accurate and illustrative techniques in determining to approximate-analytical solutions for the PDEs of fractional order in conformable sense

Approximate analytical solutions of fractional cable equation with conformable derivative operator

Bu tezde zaman-kesirli bir boyutlu kablo denklemi ele alınmıştır. Kablo denkleminin çözümünde kesirli operatör olarak uyumlu türev operatörü (UTO) kullanılmıştır. UTO ile tanımlanan kesirli kablo denklemi (UKKD)'nin çözümünde kullanılacak olan yaklaşık-analitik metotlardan; Adomian ayrışım yöntemi (AAY), varyasyonel iterasyon metodu (VİM), homotopi analiz metodu (HAM), homotopi pertürbasyon metodu (HPM), modifiye homotopi pertürbasyon metodu (MHPM) ve indirgenmiş diferansiyel dönüşüm metodu (İDDM) üzerinde durulmuştur. Bu çalışmanın asıl amacı, literatürde var olan ve bahsi geçen yaklaşık-analitik metotları UTO ile yeniden tanımlayıp bu metotlarla UKKD'nin yaklaşık-analitik çözümlerini bulmaktır. Ayrıca uyumlu türev operatörünün 2014 yılında tanımlanmış olmasından dolayı bu alanda yeterince çalışma yoktur. Bu tez çalışmasıyla birlikte UTO ile ilgili yeni bir uygulama literatüre girecektir.In this thesis, time-fractional one dimensional cable equation has been considered. Conformable derivative operator (CDO) has been used as a fractional operator in cable equation. Adomian decomposition method (ADM), variational iteration method (VIM), homotopy analysis method (HAM), homotopy perturbation method (HPM), modified homotopy perturbation method (MHPM) and reduced differential transform method (RDTM) have been emphasized in the solution of the conformable fractional cable equation (CFCE). The main aim of this study is to redefine the approximate-analytical methods that are mentioned above with CDO and to find the approximate-analytical solutions of CFCE with these suggested methods. Furhermore, since the conformable derivative operator had been defined in 2014, there are a little bit studies in this area. Therefore, a new application of CDO has been brought to the literature with this thesis