Şahin, Yasin; Merdan, Mehmet

This work investigates utilizing Adomian decomposition technique (A-ADM) coupled with Aboodh transform as a hybrid approach the solution of Caputo-Fabrizio (CF) fractional order Swift-Hohenberg problem. Effective framework for modeling memory effects and complex system dynamics is provided by CF fractional derivative defined by non-singular exponential kernel. Classical Swift-Hohenberg equation is expanded to account for long-term memory effects, which are crucial in understanding pattern development and chaotic behavior in many physical systems by including CF fractional derivatives. With the suggested Aboodh-Adomian decomposition technique (A-ADM), an approximate analytical solution is provided showing its efficiency and accuracy in solving fractional nonlinear problem. Furthermore shown are convergence and originality of the solution. Graphical depictions of the acquired solutions make use of MATLAB package software. Theoretical results are confirmed by numerical simulations, which further expose the important influence of fractional order on the dynamic features of the system.

Güngör, Nihan

Unfortunately, some of the corrections were missed in the original version and the corrections are as follows: In Theorem 1, the highlighted text should be in italics Let (Formula presented.) be (Formula presented.) continuous. Then, the following statements hold: (1) (Formula presented.) for all (Formula presented.) ,(2) (Formula presented.) for any (Formula presented.) ,(3)If (Formula presented.) for all (Formula presented.) , then (Formula presented.) ,(4)The functionfis (Formula presented.) bounded on (Formula presented.) ,(5) (Formula presented.) [10, 15]. (Formula presented.) for all (Formula presented.) , (Formula presented.) for any (Formula presented.) , If (Formula presented.) for all (Formula presented.) , then (Formula presented.) , The functionfis (Formula presented.) bounded on (Formula presented.) , (Formula presented.) [10, 15]. For (Formula presented.) , let (Formula presented.). Take f be a function with arguments in A and values in B, and let (Formula presented.). Then the following relations occur: In Theorem 9, the highlighted text should be in italics Suppose that the following conditions are satisfied:(i)f is (Formula presented.) continuous on (Formula presented.) ,(ii) (Formula presented.) is (Formula presented.) continuous on (Formula presented.) and (Formula presented.). f is (Formula presented.) continuous on (Formula presented.) , (Formula presented.) is (Formula presented.) continuous on (Formula presented.) and (Formula presented.). The Original Article was corrected.

Güngör, Nihan

In this study, the Volterra integral equations are defined in the non-Newtonian calculus with the aid of ∗ -integral. The focus of this study is obtaining the solution of linear non-Newtonian integral equations. The ∗ -resolvent kernel method, successive approximations method and Adomian decomposition method are applied for solving non-Newtonian integral equations in the non-Newtonian sense. Some numerical examples are presented to explain the procedure of these methods. Finally, a model that describes a type of the exponential growth model is proposed in the sense of non-Newtonian calculus as an example.