A Robust Study on Atangana-Baleanu Fractional Coupled Burgers Equation

Abstract

This study explores the application of the novel Atangana–Baleanu method to obtain highly accurate numerical solutions for coupled Burgers equation, which incorporates fractional derivatives in the sense of Atangana–Baleanu. To achieve this, the Atangana–Baleanu q-Elzaki homotopy analysis transform method (ABq-EHATM) is utilized as a computational technique to address the system effectively. This method uniquely combines the advantages of fractional calculus with the flexibility of q-Elzaki and homotopy analysis frameworks, offering a new perspective on handling nonlinearities and memory effects. A detailed investigation of the numerical solutions obtained through this hybrid approach is presented, emphasizing both the precision and computational efficiency of the proposed method. Furthermore, numerical simulations are carried out using Maple software for a range of fractional orders, allowing for a thorough examination of how variations in the fractional parameters influence the overall system dynamics. The results clearly demonstrate that the ABq-EHATM is not only an effective and flexible approach, but also a robust and reliable alternative for tackling complex. Overall, this study highlights a practical and innovative application of the ABq-EHATM, illustrating its potential benefits and underscoring its significance as a powerful tool in the field of fractional differential equation modeling.

Keywords

• Atangana-Baleanu Elzaki transform
• Burgers equation.
• Atangana-Baleanu q-Elzaki homotopy analysis transform method.

Atangana-Baleanu Kesirli Coupled Burgers Denklemi Üzerine Güçlü Bir Çalışma

Öz

Bu çalışma, Atangana-Baleanu anlamında kesirli türevleri içeren, bağlı Burgers denklemi için oldukça doğru sayısal çözümler elde etmek amacıyla yeni Atangana-Baleanu yönteminin uygulanmasını araştırmaktadır. Bunu başarmak için, Atangana-Baleanu q-Elzaki homotopi analiz dönüşümü yöntemi (ABq-EHATM), sistemi etkili bir şekilde ele almak için bir hesaplama tekniği olarak kullanılmıştır. Bu yöntem, kesirli hesaplamanın avantajlarını q-Elzaki homotopi analiz çerçevelerinin esnekliğiyle benzersiz bir şekilde birleştirerek, doğrusal olmayanlıkların ve bellek etkilerinin ele alınması konusunda yeni bir bakış açısı sunmaktadır. Bu hibrit yaklaşımla elde edilen sayısal çözümlerin ayrıntılı bir incelemesi sunulmakta ve önerilen yöntemin hem hassasiyeti hem de hesaplama verimliliği vurgulanmaktadır. Ayrıca, kesirli mertebeler aralığı için Maple yazılımı kullanılarak sayısal simülasyonlar gerçekleştirilmekte ve bu da kesirli parametrelerdeki değişimlerin genel sistem dinamiklerini nasıl etkilediğinin kapsamlı bir şekilde incelenmesine olanak sağlamaktadır. Sonuçlar ABqEHATM'nin yalnızca etkili ve esnek bir yaklaşım değil, aynı zamanda karmaşık doğrusal olmayan kesirli diferansiyel denklemlerle başa çıkmak için sağlam ve güvenilir bir alternatif olduğunu açıkça göstermektedir. Genel olarak, bu çalışma ABq-EHATM'nin pratik ve yenilikçi bir uygulamasını vurgulayarak potansiyel faydalarını göstermekte ve kesirli diferansiyel denklem modellemesi alanında güçlü bir araç olarak önemini vurgulamaktadır.

Anahtar Kelimeler

• Atangana-Baleanu Elzaki dönüşümü
• Burgers denklemi
• Atangana-Baleanu q-Elzaki homotopi analiz dönüşümü yöntemi

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