Alkan, Aslı; Avit, Özkan; Anaç, Halil

This study employs two innovative techniques, namely the conformable q-homotopy analysis transform method (Cq-HATM) and the conformable Elzaki Adomian decomposition method (CEADM), to investigate the numerical solutions for the conformable time-fractional coupled Whitham-Broer-Kaup problem. One of the two novel approaches offered is Cq-HATM, which is a hybrid approach that integrates the q-homotopy analysis transform method with Laplace transform using the concept of conformable derivative. The CEADM method, like the aforementioned approach, is a hybrid method that integrates the Adomian decomposition method and Elzaki transform by utilizing the concept of conformable derivative. The computer simulations were conducted in order to provide empirical evidence supporting the efficacy and reliability of the recommended methodologies. Upon comparing the precise solutions with the obtained solutions, it becomes evident that both of the novel methodologies exhibit simplicity, efficacy, and proficiency in addressing nonlinear conformable time-fractional partial differential equations.

Anaç, Halil; Avit, Özkan

This study focuses on the novel conformable methods employed to obtain new numerical solutions for the Cahn-Allen equation with conformable fractional derivatives. One of the two distinct methods put forth is the Cq-HATM, a hybrid technique that integrates the q-homotopy analysis transform method with the Laplace transform, utilizing the definition of conformable derivative. The CHPETM is a hybrid technique that combines the homotopy perturbation method with the Elzaki transform (ET). New numerical solutions of the conformal fractional differential Cahn-Allen equation were obtained using CHPETM and Cq-HATM. The computer simulations have been conducted in order to provide validation for the efficacy and reliability of the proposed methods. Upon performing a comparative analysis between the exact solutions and the solutions obtained through the novel methods, it becomes evident that both of these approaches exhibit simplicity, efficacy, and proficiency in addressing nonlinear conformable time-fractional coupled systems.

Anaç, Halil; Avit, Özkan

This research utilizes two novel methods, specifically the conformable q-homotopy analysis transform method (Cq-HATM) and the conformable Elzaki Adomian decomposition method (CEADM), to examine the numerical solutions for the conformable time-fractional coupled Jaulent-Miodek system. One of the two unique methods proposed is the Cq-HATM, which is a hybrid approach that combines the q-homotopy analysis transform method with the Laplace transform, employing the concept of conformable derivative. The CEADM method, similar to the aforementioned approach, is a hybrid technique that combines the Adomian decomposition method with Elzaki transform through the utilization of the concept of conformable derivative. The computer simulations were performed to offer validation for the effectiveness and dependability of the suggested approaches. After conducting a comparison between the exact solutions and the solutions acquired using the unique methods, it is apparent that both of these approaches demonstrate simplicity, effectiveness, and competency in tackling nonlinear conformable time-fractional coupled systems.

Avit, Özkan; Anac, Halil

The present study used the novel conformable fractional methods to examine the nonlinear Kuramoto-Sivashinsky equations of the conformable fractional derivative. The Kuramoto-Sivashinsky equation is a mathematical model employed to elucidate a diverse range of chemical and physical phenomena, encompassing chemical reaction-diffusion processes, plasma instabilities, viscous flow issues, flame front propagation, and magnetized plasmas. The present inquiry provides an examination of convergence and error for the future scheme. The conformable q-homotopy analysis transform method (Cq-HATM) provides h-curves that illustrate the convergence range of the series solution achieved. In order to validate the effectiveness and suitability of the Cq-HATM, we examine separate instances. In this study, we introduce an application that demonstrates the potential benefits and effectiveness of the proposed approach. Furthermore, an error analysis is performed in order to verify the accuracy of the scheme. Numerical simulations are conducted to validate the precision of the forthcoming approach. The findings obtained from the numerical and graphical analyses are presented in this study. The method provided in this study showcases a notable degree of computational precision and ease in examining and resolving intricate phenomena related to conformable fractional nonlinear partial differential equations within the domains of science and technology.

Özkan Avit

Bu tez,beş ana bölümden oluşmaktadır. Birinci bölümde kesirli türevin ve uyumlu kesirli türevin oluşumu ve gelişimi hakkında bilgi verilmektedir. İkinci bölümde gerekli tanım ve teoremler yazılmıştır. Üçüncü bölümde kullanılan yöntemlerden bahsedilmiştir. Dördüncü bölümde ise uyumlu kesirli mertebeden Kuramoto-Sivashinsky (UKMKS) denklemleri, uyumlu kesirli mertebeden Whitham-Broer-Kaup (UKMWBK) denklemleri, uyumlu zaman-kesirli mertebeden Jaulent Miodek (UZKMJM) sistemi, uyumlu kesirli mertebeden Cahn-Allen (UKMCA) denklemi için uyumlu q-homotopi analiz dönüşüm yöntemi (Uq-HADY), uyumlu kesirli Elzaki ayrıştırma yöntemi (UKEAY), uyumlu homotopi pertürbasyon Elzaki dönüşüm yöntemi (UHPEDM) olarak adlandırılan üç farklı yöntem kullanılarak yeni sayısal çözümler araştırılmıştır. Bu yöntemlerin etkili ve güvenilir olduğunu doğrulamak için sayısal simülasyonlar yapılmıştır. Elde edilen çözümler ile analitik çözümler arasındaki karşılaştırmalar yapılmıştır. Son bölümde ise yeni önerilen her üç yeni tekniğin de doğrusal olmayan uyumlu kesirli problemleri çözmek için basit, güçlü ve etkili olduğu vurgulanmaktadır.