Homotopi pertürbasyon Elzaki dönüşümü yöntemi ile doğrusal olmayan zaman-kesirli kısmi diferansiyel denklemler için yeni yaklaşık analitik çözümler

Öz

Bazı doğrusal olmayan zaman-kesirli mertebeden kısmi diferansiyel denklemler, homotopi pertürbasyon Elzaki dönüşümü yöntemi ile çözülmüştür. Kesirli türevler Caputo anlamında tanımlanmıştır. Uygulamalar homotopi pertürbasyon Elzaki dönüşümü yöntemi ile incelenmiştir. Bunun yanında, çözümlerin grafikleri MAPLE yazılımında çizdirilmiştir. Ayrıca homotopi pertürbasyon Elzaki dönüşümü yöntemi ve homotopi pertürbasyon Sumudu dönüşümü yöntemi çözümlerinin, lineer olmayan zaman-kesirli mertebeden kısmi diferansiyel denklemlerin tam çözümü ile mutlak hata karşılaştırması sunulmaktadır. Ek olarak, bu mutlak hata karşılaştırması tablolarda belirtilmiştir. Bu makalenin yeniliği, hem Caputo kesir dereceli gaz dinamiği denkleminin hem de Caputo kesir dereceli Klein-Gordon denkleminin bu yöntemle ilk analizidir. Bu nedenle, homotopi pertürbasyon Elzaki dönüşümü yöntemi, zaman-kesirli mertebeden kısmi diferansiyel denklemlerin analitik çözümlerinin elde edilmesinde hızlı ve etkilidir.

Anahtar Kelimeler

• Klein-gordon denklemi
• homotopi pertürbasyon elzaki dönüşüm metodu
• mittag-leffler fonksiyonu
• caputo kesirli türevi

New approximate-analytical solutions to nonlinear time-fractional partial differential equations via homotopy perturbation Elzaki transform method

Öz

Some nonlinear time-fractional partial differential equations are solved by homotopy perturbation Elzaki transform method. The fractional derivatives are defined in the Caputo sense. The applications are examined by homotopy perturbation Elzaki transform method. Besides, the graphs of the solutions are plotted in the MAPLE software. Also, absolute error comparison of homotopy perturbation Elzaki transform method and homotopy perturbation Sumudu transform method solutions with the exact solution of nonlinear time-fractional partial differential equations is presented. In addition, this absolute error comparison is indicated in the tables. The novelty of this article is the first analysis of both the gas dynamics equation of Caputo fractional order and the Klein-Gordon equation of Caputo fractional order via this method. Thus, homotopy perturbation Elzaki transform method is quick and effective in obtaining the analytical solutions of time-fractional partial differential equations.

Anahtar Kelimeler

• Klein-gordon equation
• homotopy perturbation elzaki transform method
• mittag-leffler function
• caputo fractional derivative

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