Homotopy methods for fractional linear/nonlinear differential equations with a local derivative operator

Abstract

In this paper, we consider some linear/nonlinear differential equations (DEs) containing conformable derivative operator. We obtain approximate solutions of these mentioned DEs in the form of infinite series which converges rapidly to their exact values by using and homotopy analysis method (HAM) and modified homotopy perturbation method (MHPM). Using the conformable operator in solutions of different types of DEs makes the solution steps are computable easily. Especially, the conformable operator has been used in modelling DEs and identifying particular problems such as biological, engineering, economic sciences and other some important fields of application. In this context, the aim of this study is to solve some illustrative linear/nonlinear problems as mathematically and to compare the exact solutions with the obtained solutions by considering some plots. Moreover, it is an aim to show the authenticity, applicability, and suitability of the methods constructed with the conformable operator.

Keywords

• Approximate solution,conformable operator,homotopy analysis method,modified homotopy perturbation method,nonlinear differential equations

Lokal türev operatörlü lineer/lineer olmayan diferansiyel denklemler için homotopi metotları

Abstract

Bu çalışmada conformable (uyumlu) türev operatörü (CTO) içeren bazı lineer/lineer olmayan diferansiyel denklemler ele alınmıştır. Homotopi analiz metodunu (HAM) ve modifiyeli homotopi pertürbasyon metodunu (MHPM) kullanarak bu bahsi geçen denklemlerin sonsuz seri formunda yaklaşık çözümleri elde edilmiştir. CTO kullanılması farklı türden diferansiyel denklemlerin çözümlerini elde etmede çözüm adımlarının kolay bir şekilde hesaplanmasını sağlamaktadır. Özellikle CTO mühendislik, fiziksel bilimler, ekonomi ve diğer bazı alanlardaki problemleri modellemede kullanılmaktadır. Bu bağlamda, bu çalışmanın amacı bazı lineer/lineer olmayan diferansiyel denklemleri matematiksel olarak çözmek ve çözüm grafiklerini kullanarak elde edilen yaklaşık çözümler ile tam çözümleri karşılaştırmaktır. Ayrıca CTO ile yeniden tanımlanan HAM ve MHPM metotlarının güvenirliğini, uygulanabilirliğini ve elverişliliğini göstermektir.

Keywords

• Yaklaşık çözüm,uyumlu operatör,homotopi analiz metodu,modifiyeli homotopi pertürbasyon metodu,lineer olmayan diferansiyel denklemler

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