non-Newtonian Calculus Research Articles

Convexity Properties in Non-Newtonian Calculus and Their Applications

The study presented some results on convexity properties in non-Newtonian calculus. Also presented is the Jensen-Steffensen inequality in non-Newtonian calculus and some applications. The research was mainly on positive real numbers.

An anageometric time scale calculus and its some basic applications

Anageometric calculus is a branch of non-Newtonian calculus introduced by M. Grossman and R. Katz in 1967, in which changes in functional values are measured by linear differences, but arguments are measured by ratios. The theory of time scales was initiated by S. Hilger in 1988 to unify continuous and discrete analysis. In this article, we introduce anageometric time scale calculus to discuss dynamic equations in which the values of the independent variables show nonlinear behaviour. This newly introduced calculus will be more applicable than other calculi in atomic reactors, growth models, etc. We shall discuss some applications of anageometric time scale calculus on the geometric real field R(G) with its optimality and better accuracy.

Non-Newtonian Pell and Pell-Lucas numbers

In the present paper, we introduce a new type of Pell and Pell-Lucas numbers in terms of non-Newtonian calculus, which we call non-Newtonian Pell and non-Newtonian Pell-Lucas numbers, respectively. In non-Newtonian calculus, we study some significant identities and formulas for classical Pell and Pell-Lucas numbers. Therefore, we derive some relations with non-Newtonian Pell and Pell-Lucas numbers. Furthermore, we investigate some properties of non-Newtonian Pell and Pell-Lucas numbers, including Catalan-like identities, Cassini-like identities, Binet-like formulas, and generating functions.

A Generalization of Hermite–Hadamard–Fejer Type Inequalities for the p -Convex Function via α -Generator

In the 17 t h century, I. Newton and G. Leibniz found independently each other the basic operations of calculus, i.e., differentiation and integration. And this development broke new ground in mathematics. From 1967 to 1970, Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, converting the roles of subtraction and addition into division and multiplication, respectively. And then they generalised this operation. Later, they named this analysis non-Newtonian calculus. This calculus is basically generated by generators. So, in this article, first, we give the definition of the p -convex function due to α -generator. Second, we obtain some new theorems for this function with respect to the α -generator. Third, we get some new theorems using Hermite–Hadamard–Fejer inequality for the α p -convex function. Finally, we show that our obtained results are reduced to the classical case in the special conditions.

Multiplicatively Simpson Type Inequalities via Fractional Integral

Multiplicative calculus, also called non-Newtonian calculus, represents an alternative approach to the usual calculus of Newton (1643–1727) and Leibniz (1646–1716). This type of calculus was first introduced by Grossman and Katz and it provides a defined calculation, from the start, for positive real numbers only. In this investigation, we propose to study symmetrical fractional multiplicative inequalities of the Simpson type. For this, we first establish a new fractional identity for multiplicatively differentiable functions. Based on that identity, we derive new Simpson-type inequalities for multiplicatively convex functions via fractional integral operators. We finish the study by providing some applications to analytic inequalities.

Bell-Type Inequalities from the Perspective of Non-Newtonian Calculus

AbstractA class of quantum probabilities is reformulated in terms of non-Newtonian calculus and projective arithmetic. The model generalizes spin-1/2 singlet state probabilities discussed in Czachor (Acta Physica Polonica:139 70–83, 2021) to arbitrary spins s. For $$s\rightarrow \infty$$ s → ∞ the formalism reduces to ordinary arithmetic and calculus. Accordingly, the limit “non-Newtonian to Newtonian” becomes analogous to the classical limit of a quantum theory.

On Fixed Point Results for Generalized Contractions in Non-Newtonian Metric Spaces

TThe work of non-Newtonian calculus was begun in 1972. This calculus provides a different area to the classical one. Non-Newtonian metric concept was defined in 2002 by Basar and Cakmak. Then Binbaşıoğlu et al. had given the metric spaces of non-Newtonian in 2016. Also, they started to the fixed-point theory by defining some topological properties in non-Newtonian metric spaces. In this work, we give some fixed-point theorems and results for self-mappings satisfying certain conditions in the non-Newtonian metric spaces.

A Bi-Geometric Fractional Model for the Treatment of Cancer Using Radiotherapy

Our study is based on the modification of a well-known predator-prey equation, or the Lotka–Volterra competition model. That is, a system of differential equations was established for the population of healthy and cancerous cells within the tumor tissue of a patient struggling with cancer. Besides, fractional differentiation remedies the situation by obtaining a meticulous model with more flexible parameters. Furthermore, a specific type of non-Newtonian calculus, bi-geometric calculus, can describe the model in terms of proportions and implies the alternative aspect of a dynamic system. Moreover, fractional operators in bi-geometric calculus are formulated in terms of Hadamard fractional operators. In this article, the development of fractional operators in non-Newtonian calculus was investigated. The model was extended in these criteria, and the existence and uniqueness of the model were considered and guaranteed in the first step by applying the Arzelà–Ascoli. The bi-geometric analogue of the numerical method provided a suitable tool to solve the model approximately. In the end, the visual graphs were obtained by using the MATLAB program.

A non-Newtonian Noether's symmetry theorem

The universal principle obtained by Emmy Noether in 1918 asserts that the invariance of a variational problem with respect to a one-parameter family of symmetry transformations implies the existence of a conserved quantity along with the Euler–Lagrange extremals. Here, we prove Noether's theorem for the recent non-Newtonian calculus of variations. The proof is based on a new necessary optimality condition of DuBois–Reymond type.

A New Type of Sturm-Liouville Equation in the Non-Newtonian Calculus

In mathematical physics (such as the one-dimensional time-independent Schrödinger equation), Sturm-Liouville problems occur very frequently. We construct, with a different perspective, a Sturm-Liouville problem in multiplicative calculus by some algebraic structures. Then, some asymptotic estimates for eigenfunctions of the multiplicative Sturm-Liouville problem are obtained by some techniques. Finally, some basic spectral properties of this multiplicative problem are examined in detail.

On *-Continuity and *-Uniform Continuity of Some Non-Newtonian Superposition Operators

Many studies have been done on superposition operators and non-Newtonian calculus from past to present. Sağır and Erdoğan defined Non-Newtonian superposition operators and characterized them on some sequence spaces. Also they examined *- boundedness and *-locally boundedness of Non-Newtonian superposition operators c0,α and cα to l1,β. In this study, we define *-continuity and *-uniform continuity of operator. We have proved that the necessary and sufficient conditions for the *-continuity of the non-Newtonian superposition operator c0,α to l1,β. Then we examined the relationship between the *-uniform continuity and the *-boundedness of the non-Newtonian superposition operator. Also, the similar results have been researched for the Non-Newtonian superposition operator cα to l1,β.

ON *-BOUNDEDNESS AND *-LOCAL BOUNDEDNESS OF NON-NEWTONIAN SUPERPOSITION OPERATORS IN c_(0,α) AND c_α TO l_(1,β)

Many investigations have been made about of Non-Newtonian calculus and superposition operators until today. Non-Newtonian superposition operator was defined by Sağır and Erdoğan in [9]. In this study, we have defined *- boundedness and *-locally boundedness of operator. We have proved that the non-Newtonian superposition operator $_{N}P_{f}:c_{_{0,\alpha }}\rightarrow \ell _{1,\beta }$ is *-locally bounded if and only if f satisfies the condition (NA₂′). Then we have shown that the necessary and sufficient conditions for the *-boundedness of $% _{N}P_{f}:c_{_{0,\alpha }}\rightarrow \ell _{1,\beta }$ . Finally, the similar results have been also obtained for $_{N}P_{f}:c_{\alpha }\rightarrow \ell _{1,\beta }$ .

On a Non-Newtonian Calculus of Variations

The calculus of variations is a field of mathematical analysis born in 1687 with Newton’s problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler–Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton’s approach to calculus. As in many applications negative values of admissible functions are not physically plausible, we propose here to develop an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 and 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and it is based on a (non-Newtonian) derivative that permits one to compare relative changes between a dependent positive variable and an independent variable that is also positive. In this way, the non-Newtonian calculus of variations we introduce here provides a natural framework for problems involving functions with positive images. Our main result is a first-order optimality condition of Euler–Lagrange type. The new calculus of variations complements the standard one in a nontrivial/multiplicative way, guaranteeing that the solution remains in the physically admissible positive range. An illustrative example is given.

Alternative fractional derivative operator on non-newtonian calculus and its approaches

Nowadays, study on fractional derivative and integral operators is one of the hot topics of mathematics and lots of investigations and studies make their attentions in this field. Most of these concerns raised from the vast application of these operators in study of phenomena’s models. These operators interpreted by Newtonian calculus, however different types of calculi are existed and we introduce the fractional derivative operators focused on Bi-geometric calculus and also their fractional differential equations are studied.

Some applications of extended calculus to non-Newtonian flow in pipes

Fractional and non-Newtonian calculus are an extension of classical calculus, usually known for providing new mathematical tools useful in science, developed from alternative approaches. Among fractional calculus, Riemann-Liouville and Caputo fractional derivatives have been the most popular operators employed in spite of their complexity. In this work, two novel and compact methods are presented as an alternative to the fractional calculation options. To test the feasibility of proposed methods, three classical fluid mechanic problems are studied: the flow through circular pipe, parallel plates and annulus, by modifying the constitutive equations into their fractional equivalent. On the other hand, a new weighted non-Newtonian derivative is proposed to extend the possibilities to model fluid viscosity based on the influence of nonadjacent layers, using the pipe flow as an example. Results show that proposed fractional models can describe shear-thinning and shear-thickening behaviors depending on the fractional order of the derivative, while the weighted derivative allows to expand the way viscosity is modeled, demonstrating the suitability of these approaches to describe physical problems.

Some inequalities in quasi-Banach algebra of non-Newtonian bicomplex numbers

In this paper, we construct the quasi-Banach algebra BC(N) of non-Newtonian bicomplex numbers and we generalize some topological concepts and inequalities as Schwarz?s, H?lder?s and Minkowski?s in the set of bicomplex numbers in the sense of non-Newtonian calculus.

Unifying Aspects of Generalized Calculus.

Non-Newtonian calculus naturally unifies various ideas that have occurred over the years in the field of generalized thermostatistics, or in the borderland between classical and quantum information theory. The formalism, being very general, is as simple as the calculus we know from undergraduate courses of mathematics. Its theoretical potential is huge, and yet it remains unknown or unappreciated.

Some basic properties of bigeometric calculus and its applications in numerical analysis

Objective of this paper is to visualize a new type of number line which will be called geometric number line based on geometric arithmetic introduced by Grossman and Katz in the book Non-Newtonian calculus (Grossman and Katz, Lee Press, Massachusetts, 1972). Then, we introduce different types of interpolation formulae for geometrically equal and geometrically unequal values of the argument. In our whole discussion, we’ll use geometric increment to the arguments instead of linear increments. We will apply the new formule to different problems of numerical analysis.

Non-Newtonian Mathematics Instead of Non-Newtonian Physics: Dark Matter and Dark Energy from a Mismatch of Arithmetics

Newtonian physics is based on Newtonian calculus applied to Newtonian dynamics. New paradigms such as ‘modified Newtonian dynamics’ (MOND) change the dynamics, but do not alter the calculus.However, calculus is dependent on arithmetic, that is the ways we add and multiply numbers. For example, in special relativity we add and subtract velocities by means of addition beta _1oplus beta _2=tanh big (tanh ^{-1}(beta _1)+tanh ^{-1}(beta _2)big ), although multiplication beta _1odot beta _2=tanh big (tanh ^{-1}(beta _1)cdot tanh ^{-1}(beta _2)big ), and division beta _1oslash beta _2=tanh big (tanh ^{-1}(beta _1)/tanh ^{-1}(beta _2)big ) do not seem to appear in the literature. The map f_{mathbb{X}}(beta )=tanh ^{-1}(beta ) defines an isomorphism of the arithmetic in {mathbb{X}}=(-1,1) with the standard one in {mathbb{R}}. The new arithmetic is projective and non-Diophantine in the sense of Burgin (Uspekhi Matematicheskich Nauk 32:209–210 (in Russian), 1977), while ultrarelativistic velocities are super-large in the sense of Kolmogorov (Technika Molodezhi 10:16–19 (11:30–33 in Russian), 1961). Velocity of light plays a role of non-Diophantine infinity. The new arithmetic allows us to define the corresponding derivative and integral, and thus a new calculus which is non-Newtonian in the sense of Grossman and Katz (Non-Newtonian calculus, Lee Press, Pigeon Cove, 1972). Treating the above example as a paradigm, we ask what can be said about the set {mathbb{X}} of ‘real numbers’, and the isomorphism f_{{mathbb{X}}}:{mathbb{X}}rightarrow {mathbb{R}}, if we assume the standard form of Newtonian mechanics and general relativity (formulated by means of the new calculus) but demand agreement with astrophysical observations. It turns out that the observable accelerated expansion of the Universe can be reconstructed with zero cosmological constant if f_{mathbb{X}}(t/t_H)approx 0.8sinh (t-t_1)/(0.8, t_H). The resulting non-Newtonian model is exactly equivalent to the standard Newtonian one with Omega _Lambda =0.7, Omega _M=0.3. Asymptotically flat rotation curves are obtained if ‘zero’, the neutral element 0_{{mathbb{X}}} of addition, is nonzero from the point of view of the standard arithmetic of {mathbb{R}}. This implies f^{-1}_{{mathbb{X}}}(0)=0_{{mathbb{X}}}>0. The opposition Diophantine versus non-Diophantine, or Newtonian versus non-Newtonian, is an arithmetic analogue of Euclidean versus non-Euclidean in geometry. We do not yet know if the proposed generalization ultimately removes any need of dark matter, but it will certainly change estimates of its parameters. Physics of the dark universe seems to be both geometry and arithmetic.

Modifying an approximation process using non-Newtonian calculus

In the present note we modify a linear positive Markov process of discrete type by using so called multiplicative calculus. In this framework, a convergence property and the error of approximation are established.