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- MakaleAçık Erişim2026A Non-Newtonian Extension of Laplace–Sumudu–Elzaki Transforms
Yalcin, Numan
Classical Laplace-, Sumudu-, and Elzaki-type transforms are formulated within additive analytical frameworks and do not naturally accommodate multiplicative scaling structures arising in non-Newtonian calculus. Motivated by this limitation, this study introduces a non-Newtonian Laplace–Sumudu–Elzaki transform (NNLSET) based on logarithmic scaling mechanisms, multiplicative measures, and power-type kernels. The proposed framework is constructed by replacing the classical measure dt with the multiplicative measure (Formula presented.) and the linear scaling structure (Formula presented.) with the nonlinear scaling structure (Formula presented.). Using the logarithmic transformation (Formula presented.), a canonical kernel representation of the form (Formula presented.) is derived, establishing a correspondence between multiplicative power-type kernels and weighted exponential structures in the logarithmic domain. Within an admissible weighted function framework, several analytical properties of the transform are established, including existence, boundedness, stability, uniqueness, restricted recoverability, and a logarithmic derivative representation associated with expressions of the form (Formula presented.). A comparative analysis with the classical Laplace–Sumudu–Elzaki framework, together with illustrative differential-equation examples, a representative nonlinear MEMS oscillator, and a numerical computation, is presented. The obtained results demonstrate that the NNLSET provides a mathematically consistent framework for the analysis of multiplicative structures, logarithmic scaling phenomena, and logarithmically structured differential equations. Its applicability is further illustrated through the analysis of a representative nonlinear MEMS oscillator.
- MakaleYalnız künye2024Non-Newtonian Sumudu transform
Gungor, Nihan; Dinc, Nagihan
A novel version of the Sumudu transform is introduced by using the notion of ∗-integral, which is called non-Newtonian Sumudu transform. The fundamental characteristics of the non-Newtonian Sumudu transform are demonstrated. The obtained results are applied to the solving of non-Newtonian differential equations and growth models.
- MakaleYalnız künye2022A note on linear non-Newtonian Volterra integral equations
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.
- MakaleAçık Erişim2013Spaces of continuous and bounded functions over the field of geometric complex numbers
Cakir, Zafer
Following Grossman and Katz (Non-Newtonian Calculus, 1972), we construct the sets B(A) and C(A) of geometric complex-valued bounded and continuous functions, where A denotes the compact subset of the complex plane C. We show that the sets B(A) and C(A) of complex-valued bounded and continuous functions form a vector space with respect to the addition and scalar multiplication in the sense of multiplicative calculus. Finally, we prove that B(A) and C(A) are complete metric spaces.