Numerical Solutions of Conformable Partial Differential Equations By Homotopy Analysis Method

In this paper, the Homotopy Analysis Method (HAM) is applied to the fractional Wu-Zhang system and combined KdV-mKdV equation to obtain theirnumerical solutions. The results were compared with analytical solutions.Bu makalede, kesirli Wu-Zhang sisteminin ve birleştirilmiş KdV-mKdV denklemidenkleminin nümerik çözümlerinielde etmek için Homotopi Analiz Yöntemi (HAM) uygulandı. Elde edilen sonuçlar, analitik çözümlerler ile karşılaştırıldı

Solving fractional diffusion and fractional diffusion-wave equations by Petrov-Galerkin finite element method

In the last few years, it has become highly evident that fractional calculus has been widely used in several areas of science. Because of this fact, their numerical solutions also have become urgently important. In this manuscript, numerical solutions of both the fractional diffusion and fractional diffusion-wave equations have been obtained by a Petrov-Galerkin finite element method using quadratic B-spline base functions as trial functions and linear B-spline base functions as the test functions. In those equations, fractional derivatives are used in terms of the Caputo sense. While the L1 discretizaton formula has been applied to fractional diffusion equation, the L2 discretizaton formula has been applied to the fractional diffusion-wave equation. Finally, the error norms L2 and L∞ have been calculated for testing the accuracy of the proposed scheme.Publisher's Versio

The exact solutions of conformable fractional partial differential Equations using new sub equation method

In this article, authors employed the new sub equation method to attain new traveling wave solutions of conformable time fractional partial differential equations. Conformable fractional derivative is a well behaved, applicable and understandable definition of arbitrary order derivation. Also this derivative obeys the basic properties that Newtonian concept satisfies. In this study authors obtained the exact solution for KDV equation where the fractional derivative is in conformable sense. New solutions are obtained in terms of the generalized version of the trigonometric functions

The New Travelling Wave Solutions of Time Fractional Fitzhugh-Nagumo Equation with Sine-Gordon Expansion Method

Authors aimed to employ the sine-Gordon expansion method to acquire the new exact solutions of fractional Fitzhugh-Nagumo equation which is a stripped type of the Hodgkin-Huxley model that expresses in extensive way activation and deactivation dynamics of neuron spiking. By using the wave transformations, by the practicality of chain rule and applicability of the conformable fractional derivative, the fractional nonlinear partial differential equation (FNPDE) changes to a nonlinear ordinary differential equation. So the exact solution of the considered equation can be obtained correctly with the aid of efficient and reliable analytical techniques. Keywords: Sine-Gordon expansion method; Fitzhugh-Nagumo Equation; Conformable derivative

Analytical solutions of fractional order partial diferantial equations with the help of auxiulary equation method

Bu makalede conformable kesirli mertebeden türev içeren lineer olmayan kesirli mertebeden Konopelchenko-Dubrovsky ve Benjamin-Ono denklemlerinin analitik çözümleri yardımcı denklem yöntemi ile elde edildi. Her iki denklem dalga dönüşümü yardımıyla lineer olmayan adi türevli diferansiyel denkleme dönüştürüldükten sonra yardımcı denklem yöntemi kullanıldı. Elde edilen çözümlerin üç boyutlu grafikleri verildi. Yardımcı denklem yöntemi, diğer analitik yöntemlere göre daha çok analitik çözüm içermektedir.In this article, the analytical solutions of nonlinear fractional order Konopelchenko-Dubrovsky and Benjamin-Ono equations are obtained with the aid of auxiulary equation method where the fractional derivatives are in conformable sense. After both equations were converted to nonlinear ordinary differential equations with the help of wave transformation, auxiliary equation method was used. Three dimensional graphics of the obtained results are given. The auxiliary equation method contains more analytical solutions than other analytical methods

New Exact Solutions of Fractional Fitzhugh-Nagumo Equation

The main aim of this article is obtaining the travelling wave, solitary wave and periodicwave solutions for time fractional Fitzhugh-Nagumo equation which used as a model for reaction–diffusion, transmission of nerve impulses, circuit theory, biology and population genetics. The newextended direct algebraic method is employed for this aim. The fractional derivative is in conformablesense which is an applicable, well behaved and understandable definition

Zaman-kesirli Kadomtsev- Petviashvili denkleminin conformable türev ile yaklaşık çözümleri

In this study, residual power series method, namely RPSM, is applied to solve time-fractional KadomtsevPetviashvili (K-P) differential equation. In the solution procedure, the fractional derivatives are explained in the conformable sense. The model is solved approximately and the obtained results are compared with exact solutions obtained by the sub-equation method. The results reveal that the present method is accurate, dependable, simple to apply and a good alternative for seeking solutions of nonlinear fractional partial differential equations.Bu çalışmada, zaman-kesirli Kadomtsev-Petviashvili (K-P) diferansiyel denklemini çözmek için Rezidual Kuvvet Serisi Metodu (RPSM) kullanılmıştır. Çözüm prosedüründe, kesirli türevler, conformable kesirli türev tanımına göre hesaplanmıştır. Bu model yaklaşık olarak çözülmüş ve elde edilen sonuçlar, sub-equation metodu ile elde edilen tam çözümlerle karşılaştırılmıştır. Sonuçlar, mevcut yöntemin doğru, güvenilir, uygulanmasının basit olduğunu ve doğrusal olmayan kısmi diferansiyel denklemlerin çözümü için iyi bir alternatif olduğunu ortaya koymaktadır

Küçük genlikli lineer olmayan uzun dalgaların gelişiminde türeyen zaman kesirli Kadomtsev-Petviashvili denkleminin yeni dalga çözümleri

The main aim of this paper is to obtain the travelling wave solutions of fractional Kadomtsev- Petviashvili(KP) Equation where the derivative is in conformable sense. For this aim the sub equation method is used with computer software called Mathematica. Then, solutions are investigated through the graphical representation for different cases of D.Bu makalenin asıl amacı, türevleri conformable cinsinden olan kesirli Kadomtsev-Petviashvili (KP) denkleminin dalga çözümlerini bulmaktır. Bu amaç için alt denklem metodu, Mathematica bilgisayar programı ile birlikte kullanılmıştır. Daha sonra elde edilen çözümlerin D ’ nın değişik değerleri için grafiksel gösterimleri verilmiştir

The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method

In this article, authors employed the new sub equation method to attain new traveling wave solutions of conformable time fractional partial differential equations. Conformable fractional derivative is a well behaved, applicable and understandable definition of arbitrary order derivation. Also this derivative obeys the basic properties that Newtonian concept satisfies. In this study authors obtained the exact solution for KDV equation where the fractional derivative is in conformable sense. New solutions are obtained in terms of the generalized version of the trigonometric functions