Delta Derivatives Research Articles

Qualitative Outcomes on Monotone Iterative Technique and Quasilinearization Method on Time Scale

In this paper, a nonlinear dynamic equation with an initial value problem (IVP) on a time scale is considered. First, applying comparison results with a coupled lower solution (LS) and an upper solution (US), we improved the quasilinearization method (QLM) for the IVP. Unlike other studies, we consider the LS and US pair of the seventh type instead of the natural type. It was determined that the solutions of the dynamic equation converge uniformly and monotonically to the unique solution of the IVP, and the convergence is quadratic. Moreover, we will use the delta derivative (Δγ) instead of the classical derivative (dγ) in the proof because it studies a time scale. In the second part of the paper, we applied the monotone iterative technique (MIT) coupled with the LS and US, which is an effective method, proving a clear analytical representation for the solution of the equation when the relevant functions are monotonically non-decreasing and non-increasing. Then an example is given to illustrate the results obtained.

Inventive dynamic inequalities of Pachpatte type on time scales and applications

In this article, we report a set of unique dynamic inequalities for different time scales that are based on delta-derivatives. These proposed inequalities prove to be useful tools for investigating both qualitative and quantitative facets of a class of time-scale dynamic equations, in addition to enhancing and extending several inequalities found in the literature. The introduced inequalities aid in a more thorough comprehension of the features of these equations, which involve an unknown function and its associated delta derivatives. The endpoint of this article includes numerical illustrations to emphasize the practical significance of the developed results.

Diamond-Type Dirac Dynamic System in Mathematical Physics

In order to merge continuous and discrete analyses, a number of dynamic derivative equations have been put out in the process of developing a time-scale calculus. The investigations that incorporated combined dynamic derivatives have led to the proposal of improved approximation expressions for computational application. One such expression is the diamond alpha (⋄α) derivative, which is defined as a linear combination of delta and nabla derivatives. Several dynamic equations and inequalities, as well as hybrid dynamic behavior—which does not occur in the real line or on discrete time scales—are analyzed using this combined concept. In this study, we consider a ⋄α Dirac system under boundary conditions on a uniform time scale. We examined some basic spectral properties of the problem we are considering, such as the simplicity, the reality of eigenvalues, orthogonality of eigenfunctions, and self adjointness of the operator. Finally, we construct an expression for the eigenfunction of the ⋄α Dirac boundary value problem (BVP) on a uniform time scale.

Exact Treatment of the Infinite Square Well in One Dimension with λδ^' (x) Potential

Abstract: This work considered the infinite square well in one dimension with a contact potential. The Dirac delta derivative function potential λδ^' where λ is a coupling constant was used to represent the contact potential. Using Green’s function technique, exact implicit expressions of the energy eigenvalues and eigenfunctions were obtained. The energy eigenvalues were expressed using a transcendental equation. The energy eigenfunctions satisfy the Schrödinger equation and the infinite square well boundary conditions. Also, the eigenfunctions and their first derivative were shown to be discontinuous. The values of these discontinuity jumps agreed with the required conditions for a self-adjoint extension Hamiltonian. In the weak coupling region, the energy eigenvalues are close to that of the even parity solution before adding the contact potential. The energy eigenvalues in the strong coupling regime reveal the energy eigenvalues of the odd parity solution. Keywords: Point interactions, Infinite square well, Green’s function technique. PACS numbers: 03.65.-w, 03.65.Db, 03.65.Ge

On some investigations of alpha-conformable Ostrowski–Trapezoid–Grüss dynamic inequalities on time scales

We prove new Ostrowski-type α-conformable dynamic inequalities and its companion inequalities on time scales by using the integration-by-parts formula on time scales associated with two parameters for functions with bounded second delta derivatives. When alpha =1, we obtain some well-known time-scale inequalities due to Ostrowski. As particular cases, we obtain new continuous and discrete inequalities.

A Variety of Weighted Opial-Type Inequalities with Applications for Dynamic Equations on Time Scales

Using higher order delta derivatives on time scales, we demonstrated a few dynamic inequalities of the Opial type in this paper. Our findings expanded upon and generalised earlier findings in the literature. Furthermore, we give the discrete and continuous inequalities as special cases. At the end of this paper, we apply our results to study the behaviour of the solution of an initial value problem. In selecting the best ways to solve dynamic inequalities, symmetry is crucial.

On some dynamic inequalities of Ostrowski, trapezoid, and Grüss type on time scales

In the present manuscript, we prove some new extensions of the dynamic Ostrowski inequality and its companion inequalities on an arbitrary time scale by using two parameters for functions whose second delta derivatives are bounded. In addition to these generalizations, some integral and discrete inequalities are obtained as special cases of main results.

Delay dynamic equations on isolated time scales and the relevance of one‐periodic coefficients

We are motivated by the idea that certain properties of delay differential and difference equations with constant coefficients arise as a consequence of their one‐periodic nature. We apply the recently introduced definition of periodicity for arbitrary isolated time scales to linear delay dynamic equations and a class of nonlinear delay dynamic equations. Utilizing a derived identity of higher order delta derivatives and delay terms, we rewrite the considered linear and nonlinear delayed dynamic equations with one‐periodic coefficients as a linear autonomous dynamic system with constant matrix. As the simplification of a constant matrix is only obtained for one‐periodic coefficients, dynamic equations with one‐periodic coefficients are the simplest form compared to the commonly used constant coefficients.

Molecular and Epidemiological Characterization of Emerging Immune-Escape Variants of SARS-CoV-2.

The successive emergence of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) variants has presented a major challenge in the management of the coronavirus disease (COVID-19) pandemic. There are growing concerns regarding the emerging variants escaping vaccines or therapeutic neutralizing antibodies. In this study, we conducted an epidemiological survey to identify SARS-CoV-2 variants that are sporadically proliferating in vaccine-advanced countries. Subsequently, we created HiBiT-tagged virus-like particles displaying spike proteins derived from the variants to analyze the neutralizing efficacy of the BNT162b2 mRNA vaccine and several therapeutic antibodies. We found that the Mu variant and a derivative of the Delta strain with E484K and N501Y mutations significantly evaded vaccine-elicited neutralizing antibodies. This trend was also observed in the Beta and Gamma variants, although they are currently not prevalent. Although 95.2% of the vaccinees exhibited prominent neutralizing activity against the prototype strain, only 73.8 and 78.6% of the vaccinees exhibited neutralizing activity against the Mu and the Delta derivative variants, respectively. A long-term analysis showed that 88.8% of the vaccinees initially exhibited strong neutralizing activity against the currently circulating Delta strain; the number decreased to 31.6% for the individuals at 6 months after vaccination. Notably, these variants were shown to be resistant to several therapeutic antibodies. Our findings demonstrate the differential neutralization efficacy of the COVID-19 vaccine and monoclonal antibodies against circulating variants, suggesting the need for pandemic alerts and booster vaccinations against the currently prevalent variants.

Delta derivatives of the solution to a third-order parameter dependent boundary value problem on an arbitrary time scale

Delta derivatives of the solution to a third-order parameter dependent boundary value problem on an arbitrary time scale

New Fractional Dynamic Inequalities via Conformable Delta Derivative on Arbitrary Time Scales

Building on the work of Josip Pečarić in 2013 and 1982 and on the work of Srivastava in 2017. We prove some new α-conformable dynamic inequalities of Steffensen-type on time scales. In the case when α=1, we obtain some well-known time scale inequalities due to Steffensen inequalities. For some specific time scales, we further show some relevant inequalities as special cases: α-conformable integral inequalities and α-conformable discrete inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

Solving Riccati Type q−Difference Equations via Difference Transform Method

In this paper, we deal on the time scale that its delta derivative of graininess function is a nonzero positive constant. Based on the Taylor formula for this time scale, we investigate the difference transform method (DTM). This method has been applied successfully to solve Riccati type difference equations in quantum calculus. To demonstrate the ability and efficacy of this method, some examples have been provided.

MFKK Özniteliklerine Eklenen Logaritmik Enerji ve Delta Parametrelerinin Yaş ve Cinsiyet Sınıflandırma Üzerindeki Etkileri

Konuşmacıların yaş ve cinsiyet gruplarının otomatik olarak belirlenmesi önemli bir araştırma konusudur ve başta çağrı merkezleri olmak üzere birçok alanda farklı amaçlarla kullanılmaktadır. Bu çalışmada Mel Frekansı Kepstrum Katsayılarına (MFKK) eklenen logaritmik enerji ve delta parametrelerinin otomatik yaş ve cinsiyet tanıma üzerindeki etkileri araştırılmıştır. Konuşma sinyallerinden çıkarılan MFKK öznitelikleri, Gauss Karışım Modeli (GKM) süpervektörlerine dönüştürüldükten sonra Destek Vektör Makinesine (DVM) uygulanmış ve gerçekleştirilen optimizasyon süreci sonunda konuşmacıların yaş ve cinsiyet gruplarına karar verilmiştir. Çalışmada MFKK’ya eklenen parametrelerin yanı sıra MFKK sayısının ve GKM bileşen sayısının başarı üzerindeki etkileri de araştırılmıştır. MFKK sayısı 8 ile 20, GKM bileşen sayısı ise 32 ile 256 arasında değiştirilerek sistem üzerinde testler yapılmıştır. aGender veritabanının geliştirme bölümündeki 299 konuşmacının 1388 konuşması ile yapılan testlerde en yüksek sınıflandırma oranı, 12 kepstral katsayıya logaritmik enerji, delta ve delta-delta parametrelerinin eklenmesi sonucunda %60.23 olarak hesaplanmıştır. Çalışmada optimum GKM bileşen sayısı 128 olarak belirlenirken, logaritmik enerji, delta ve delta-delta parametrelerinin başarı üzerindeki etkileri sırasıyla %1.17, %3.24 ve %4.61 olarak saptanmıştır.

Exact solutions for the Lippmann–Schwinger equation in two dimensions and invisibility conditions

We present exact solutions for the Lippmann–Schwinger equation in two dimensions for circular boundary walls in three cases: (i) a finite number N of concentric barriers; (ii) a single barrier with Dirac delta derivatives, in the sense of distribution theory, namely, angular, normal, and along the curve; and (iii) a single barrier with an arbitrary distribution. As an application of this last result, we obtain conditions that must be fulfilled in order for the barrier to become invisible.

A generalized delta derivative on time scale with applications

Inspired by a well‐known “Hilger‐derivative” on time scale introduced by Stephan Hilger, we introduce a new derivative on the time scale, called “black delta (or in notation )‐derivative.” It is an adaptive unified approach of derivative of continuous and discrete calculus. Several important nontrivial results concerning this derivative are obtained. Also, a necessary and sufficient condition for the existence of this new derivative and its connection with Hilger‐derivative are discussed. In addition, a new general a‐exponential function (for fixed a ≥ 1) is introduced to obtain the unique solution of the initial value problem on time scales with this derivative. Furthermore, a relation between the a‐exponential function with the existing exponential function on the time scale is obtained. We elucidate the results by considering some interesting examples. We expect that this derivative will be applicable for future investigations on various studies in the field of mathematical modeling, engineering applications.

Xistence of positive solutions for nonlinear multipoint p-Laplacian dynamicequations on time scales

In this paper, we investigate the existence of positive solutions for nonlinear multipoint boundary value problems for p-Laplacian dynamic equations on time scales with the delta derivative of the nonlinear term. Sufficient assumptions are obtained for existence of at least twin or arbitrary even positive solutions to some boundary value problems. Our results are achieved by appealing to the fixed point theorems of Avery-Henderson. As an application, an example to demonstrate our results is given.

Some opial-type inequalities with higher order delta derivatives on time scales

In this article, we will establish some new dynamic Opial-type inequalities with higher order delta derivatives on time scales. Our results generalize some existing dynamic Opial-type inequalities, and give some integral and discrete inequalities as special cases. An application will be introduced to illustrate the benefit of some of our results.

Contact Interactions in One-Dimensional Quantum Mechanics: a Family of Generalized б'-Potentials

A “one-point” approximation is proposed to investigate the transmission of electrons through the extra thin heterostructures composed of two parallel plane layers. The typical example is the bilayer for which the squeezed potential profile is the derivative of Dirac’s delta function. The Schr¨odinger equation with this singular one-dimensional profile produces a family of contact (point) interactions each of which (called a “distributional” б′-potential) depends on the way of regularization. The discrepancies widely discussed so far in the literature regarding the family of delta derivative potentials are eliminated using a two-scale power-connecting parametrization of the bilayer potential that enables one to extend the family of distributional б′-potentials to a whole class of “generalized” б′-potentials. In a squeezed limit of the bilayer structure to zero thickness, the resonant tunneling through this structure is shown to occur in the form of sharp peaks located on the sets of Lebesgue’s measure zero (called resonance sets). A four-dimensional parameter space is introduced for the representation of these sets. The transmission on the complement sets in the parameter space is shown to be completely opaque.

Time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems.

The time-scales theory provides a powerful theoretical tool for studying differential and difference equations simultaneously. With regard to Herglotz type variational principle, this generalized variational principle can deal with non-conservative or dissipative problems. Combining the two tools, this paper aims to study time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems. We introduce the time-scales Herglotz type variational problem of Birkhoffian systems firstly and give the form of time-scales Pfaff–Herglotz action for delta derivatives. Then, time-scales Herglotz type Birkhoff’s equations for delta derivatives are derived by calculating the variation of the action. Furthermore, time-scales Herglotz type Noether symmetry for delta derivatives of Birkhoffian systems are defined. According to this definition, time-scales Herglotz type Noether identity and Noether theorem for delta derivatives of Birkhoffian systems are proposed and proved, which can become the ones for delta derivatives of Hamiltonian systems or Lagrangian systems in some special cases. Therefore, it is shown that the results of Birkhoffian formalism are more universal than Hamiltonian or Lagrangian formalism. Finally, the time-scales damped oscillator and a non-Hamiltonian Birkhoffian system are given to exemplify the superiority of the results.

Lie symmetries of the relative motion systems on time scales

In this paper, the Lie symmetries and conserved quantities of the relative motion systems on time scales are proposed and studied. The Lagrange equations with delta derivatives on time scales are derived. By defining the infinitesimal transformations generators and using the invariance of differential equations under infinitesimal transformations, the determining equations of the Lie symmetries on time scales are established. Then, the structural equation and the form of conserved quantities of the Lie symmetries are given. Lie symmetries of the relative motion systems in discrete and continuous systems were discussed, respectively. Finally, an example is given to illustrate the applications of the conclusion.