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In this article, we examine the Kraenkel–Manna–Merle system (KMMS) with an M-truncated derivative (MTD). Our goal is to obtain rational, hyperbolic, and trigonometric solutions by using the F-expansion technique with the Riccati equation. To our knowledge, no one has studied the exact solutions to the KMMS in the presence/absence of a damping effect with an M-truncated derivative, using the F-expansion technique. The magnetic field propagation in a zero-conductivity ferromagnet is described by the KMMS; hence, solutions to this equation may provide light on several fascinating scientific phenomena. We use MATLAB to display figures in a variety of 3D and 2D formats to demonstrate the influence of the M-truncated derivative on the exact solutions to the KMMS.

In this study, we extended and improved the oscillation criteria previously established for second-order differential equations to even-order differential equations. Some examples are given to demonstrate the significance of the results accomplished.

The main objective of this work is to deduce some interesting algebraic relationships that connect the degenerated generalized Apostol–Bernoulli, Apostol–Euler and Apostol– Genocchi polynomials and other families of polynomials such as the generalized Bernoulli polynomials of level m and the Genocchi polynomials. Futher, find new recurrence formulas for these three families of polynomials to study.

In this paper, we construct an analytical solution of the coupled Burgers’ equation, using the homotopy analysis method, which is a semi-analytical method, the approximate solution obtained by this method is convergent for different values of the convergence control parameter ℏ, the optimal value of ℏ corresponding with the minimum error to be determined by the residual. The results obtained by the present method are compared with other obtained solutions by different numerical methods.

This paper aims to establish some new formulas and results related to the Erdélyi-Kober fractional integral operator applied to the extended hypergeometric functions. The results are expressed as the Hadamard product of the extended and confluent hypergeometric functions. Some special cases of our main results are also derived.

For third order linear differential equations of the form r(t)x'(t)''+ p(t)x'(t) + q(t)x(t) = 0; we will establish lower bounds for the distance between zeros of a solution and/or its derivatives. The main results will be proved by making use of Hardyís inequality, some generalizations of Opialís inequality and Boydís inequality.

In this work, we study the oscillation of the fourth order neutral differential equations with delay argument. By means of generalized Riccati transformation technique, we obtain new oscillation criteria for oscillation of this equation. An example is given to clarify the main results in this paper.

The objective of this paper is to study the oscillation criteria for odd-order neutral differential equations with several delays. We establish new oscillation criteria by using Riccati transformation. Our new criteria are interested in complementing and extending some results in the literature. An example is considered to illustrate our results.

By employing the Riccati substitution technique, we establish new oscillation criteriafor a class of fourth-order neutral differential equations. Our new criteria complement a number of existing ones. An illustrative example is provided.

In this article, the variational iteration algorithm-I with an auxiliary parameter (VIA-I with AP) is elaborated to initial and boundary value problems. The effectiveness, absence of difficulty and accuracy of the proposed method is remarkable and its tractability is well suitable for the use of these type of problems. Some examples have been given to show the effectiveness and utilization of this technique. A comparison of variational iteration algorithm-I (VIA-I) along VIA-I with AP has been carried out. It can be seen that this technique is more appropriate than as VIA-I.

In this work, we study the oscillatory properties of a higher-order neutral delay differential equation. By using the principle of comparison with equations of the first order we establish a set of new oscillation criteria for this equation under the canonical condition. Furthermore, the new criteria extend and complement some previous results in the literature. To that end, we compare these criteria by applying them to special cases of the equations under consideration in order to determine which one is the most efficient and least restrictive.

In this article, we utilize recent generalized fractional operators to establish some fractional inequalities in Hermite–Hadamard and Minkowski settings. It is obvious that many previously published inequalities can be derived as particular cases from our outcomes. Moreover, we articulate some flaws in the proofs of recently affiliated formulas by revealing the weak points and introducing more rigorous proofs amending and expanding the results.

In this paper, we consider the (4+1)-dimensional fractional Fokas equation (FFE) with an M-truncated derivative. The extended tanh–coth method and the Jacobi elliptic function method are utilized to attain new hyperbolic, trigonometric, elliptic, and rational fractional solutions. In addition, we generalize some previous results. The acquired solutions are beneficial in analyzing definite intriguing physical phenomena because the FFE equation is crucial for explaining various phenomena in optics, fluid mechanics and ocean engineering. To demonstrate how the M-truncated derivative affects the analytical solutions of the FFE, we simulate our figures in MATLAB and show several 2D and 3D graphs.

The stochastic fractional-space Korteweg–de Vries equation (SFSKdVE) in the sense of the M-truncated derivative is examined in this article. In the Itô sense, the SFSKdVE is forced by multiplicative white noise. To produce new trigonometric, hyperbolic, rational, and elliptic stochastic fractional solutions, the tanh–coth and Jacobi elliptic function methods are used. The obtained solutions are useful in interpreting certain fascinating physical phenomena because the KdV equation is essential for understanding the behavior of waves in shallow water. To demonstrate how the multiplicative noise and the M-truncated derivative impact the precise solutions of the SFSKdVE, different 3D and 2D graphical representations are plotted.

The main aim of this work is to study an extension of the Caputo fractional derivative operator by use of the two-parameter Mittag–Leffler function given by Wiman. We have studied some generating relations, Mellin transforms and other relationships with extended hypergeometric functions in order to derive this extended operator. Due to symmetry in the family of special functions, it is easy to study their various properties with the extended fractional derivative operators.

In this paper, we propose some generalized integral inequalities of the Raina type depicting the Mittag–Leffler function. We introduce and explore the idea of generalized s-type convex function of Raina type. Based on this, we discuss its algebraic properties and establish the novel version of Hermite–Hadamard inequality. Furthermore, to improve our results, we explore two new equalities, and employing these we present some refinements of the Hermite–Hadamard-type inequality. A few remarkable cases are discussed, which can be seen as valuable applications. Applications of some of our presented results to special means are given as well. An endeavor is made to introduce an almost thorough rundown of references concerning the Mittag–Leffler functions and the Raina functions to make the readers acquainted with the current pattern of emerging research in various fields including Mittag–Leffler and Raina type functions. Results established in this paper can be viewed as a significant improvement of previously known results.

Here, we analyze the (2+1)-dimensional stochastic modified Kordeweg–de Vries (SmKdV) equation perturbed by multiplicative white noise in the Stratonovich sense. We apply the mapping method to obtain new trigonometric, elliptic, and rational stochastic fractional solutions. Because of the importance of the KdV equation in characterizing the behavior of waves in shallow water, the obtained solutions are beneficial in interpreting certain fascinating physical phenomena. We plot our figures in MATLAB and show several 3D and 2D graphical representations to show how the multiplicative white noise affects the solutions of the SmKdV. We show that the white noise around zero stabilizes SmKdV solutions.

Equiform geometry is considered an extension of other geometries. Furthermore, an equiform frame is a generalization of the Frenet frame. In this study, we begin by defining the term “equiform parameter (EQP)”, “equiform frame”, and “equiform formulas (EQF)” in regard to the Minkowski three-space. Second, we define spacelike normal curves (SPN) in Minkowski three-space and present a variety of descriptions of these curves with equiform spacelike (EQS) or equiform timelike (EQN) principal normals in Minkowski three-space. Third, we discuss the implications of these findings. Finally, an example is given to illustrate our theoretical results.

The main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third order with noncanonical operator (a(ι)[(b(ι)x(ι)+p(ι)x(ι−τ)′)′]β)′+∫cdq(ι,μ)xβ(σ(ι,μ))dμ=0, where ι≥ι0 and w(ι):=x(ι)+p(ι)x(ι−τ). New oscillation results are established by using the generalized Riccati technique under the assumption of ∫ι0ιa−1/β(s)ds&lt, ∫ι0ι1b(s)ds=∞asι→∞. Our new results complement the related contributions to the subject. An example is given to prove the significance of new theorem.

Differential equations with delay arguments are one of the branches of functional differential equations which take into account the system’s past, allowing for more accurate and efficient future prediction. The symmetry of the equations in terms of positive and negative solutions plays a fundamental and important role in the study of oscillation. In this paper, we study the oscillatory behavior of a class of odd-order neutral delay differential equations. We establish new sufficient conditions for all solutions of such equations to be oscillatory. The obtained results improve, simplify and complement many existing results.

Starting from the heat equation, we discuss some fractional generalizations of various forms. We propose a method useful for analytic or numerical solutions. By using Hermite polynomials of higher and fractional order, we present some operational techniques to find general solutions of extended form to d'Alembert and Fourier equations. We also show that the solutions of the generalized equations discussed here can be expressed in terms of Hermite-based functions.

Chebyshev polynomials are traditionally applied to the approximation theory where are used in polynomial interpolation and also in the study of di erential equations, in particular in some special cases of Sturm-Liouville di erential equation. Many of the operational techniques presented, by using suitable integral transforms, via a symbolic approach to the Laplace transform, allow us to introduce polynomials recognized belonging to the families of Chebyshev of multi-dimensional type. The non-standard approach come out from the theory of multi-index Hermite polynomials, in particular by using the concepts and the related formalism of translation operators.

Oscillation and symmetry play an important role in many applications such as engineering, physics, medicine, and vibration in flight. The purpose of this article is to explore the oscillation of fourth-order differential equations with delay arguments. New Kamenev-type oscillatory properties are established, which are based on a suitable Riccati method to reduce the main equation into a first-order inequality. Our new results extend and simplify existing results in the previous studies. Examples are presented in order to clarify the main results.

In this paper, we derive some new fractional extensions of Hardy’s type inequalities. The corresponding reverse relations are also obtained by using the conformable fractional calculus from which the classical integral inequalities are deduced as special cases at α=1.

Using the concepts and formalism of different families of Hermite polynomials, we discuss here some generalizations of polynomials belonging to the Bernoulli class, and we also show how to represent the action of the operators involving fractional derivatives. In particular, by using the method of generating function, we introduce generalized Bernoulli polynomials by operating in their generating function with the formalism of the two-variable Hermite polynomials. In addition, we extend some operational techniques in order to derive different forms of Bernoulli numbers and polynomials. Finally, we explore some general properties of generalized Bernoulli polynomials, focusing on their extension to the 2D case, and we introduce a family of polynomials strictly related to the Hermite polynomials in order to compute the effect of fractional operators on a given function.

Nonhomogeneous systems of fractional differential equations with pure delay are considered. As an application, the representation of solutions of these systems and their delayed Mittag-Leffler matrix functions are used to obtain the finite time stability results. Our results improve and extend the previous related results. Finally, to illustrate our theoretical results, we give an example.

The aim of this work is to improve the oscillation results for second-order neutral differential equations with damping term. We consider the noncanonical case which always leads to two independent conditions for oscillation. We are working to improve related results by simplifying the conditions, based on taking a different approach that leads to one condition. Moreover, we obtain different forms of conditions to expand the application area. An example is also given to demonstrate the applicability and strength of the obtained conditions over known ones.

The applications of fractional partial differential equations (PDEs) in diverse disciplines of science and technology have caught the attention of many researchers. This article concerned with the approximate numerical solutions of three-dimensional two- and three-term time fractional PDE models utilizing an accurate, and computationally attractive local meshless technique. Due to their tremendous advantages like ease of applicability in higher dimensions in both regular and irregular domains, the interest in meshless techniques is increasing. Test problems are considered to assess the reliability and accuracy of the proposed technique.

The main aim of this research paper is to introduce a new extension of the Gauss hypergeometric function and confluent hypergeometric function by using an extended beta function. Some functional relations, summation relations, integral representations, linear transformation formulas, and derivative formulas for these extended functions are derived. We also introduce the logarithmic convexity and some important inequalities for extended beta function.

This work is devoted to providing new kinds of deterministic and stochastic solutions of one of the famous nonlinear equations that depends on time, called the Schrödinger–Hirota equation. A new and straightforward methodology is offered to extract exact wave solutions of the stochastic nonlinear evolution equations (NEEs) with generalized differential conformable operators (GDCOs). This methodology combines the features of GDCOs, some instruments of white noise analysis, and the generalized Kudryashov scheme. To demonstrate the usefulness and validity of our methodology, we applied it to extract diversified exact wave solutions of the Schrödinger–Hirota equation, particularly in a Wick-type stochastic space and with GDCOs. These wave solutions can be turned into soliton and periodic wave solutions that play a main role in numerous fields of nonlinear physical sciences. Moreover, three-dimensional, contour, and two-dimensional graphical visualizations of some of the extracted solutions are exhibited with some elected functions and parameters. According to the results, our new approach demonstrates the impact of random and conformable factors on the solutions of the Schrödinger–Hirota equation. These findings can be applied to build new models in plasma physics, condensed matter physics, industrial studies, and optical fibers. Furthermore, to reinforce the importance of the acquired solutions, comparative aspects connected to some former works are presented for these types of solutions.

In this study, new asymptotic properties of positive solutions of the even-order delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Moreover, we use these properties to obtain new criteria for the oscillation of the solutions of the studied equation using the principles of comparison.

In this article, two new modified variational iteration algorithms are investigated for the numerical solution of coupled Burgers&prime, equations. These modifications are made with the help of auxiliary parameters to speed up the convergence rate of the series solutions. Three numerical test problems are given to judge the behavior of the modified algorithms, and error norms are used to evaluate the accuracy of the method. Numerical simulations are carried out for different values of parameters. The results are also compared with the existing methods in the literature.

We show that a Fuchsian differential equation having five regular singular points admits solutions in terms of a single generalized hypergeometric function for infinitely many particular choices of equation parameters. Each solution assumes four restrictions imposed on the parameters: two of the singularities should have non-zero integer characteristic exponents and the accessory parameters should obey polynomial equations.

In this paper, a new class of ( C , G f ) -invex functions introduce and give nontrivial numerical examples which justify exist such type of functions. Also, we construct generalized convexity definitions (such as, ( F , G f ) -invexity, C-convex etc.). We consider Mond&ndash, Weir type fractional symmetric dual programs and derive duality results under ( C , G f ) -invexity assumptions. Our results generalize several known results in the literature.

In this paper, we study the oscillation of second-order neutral differential equations with delayed arguments. Some new oscillatory criteria are obtained by a Riccati transformation. To illustrate the importance of the results, one example is also given.

In the paper, we study the oscillation of fourth-order delay differential equations, the present authors used a Riccati transformation and the comparison technique for the fourth order delay differential equation, and that was compared with the oscillation of the certain second order differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. Some examples are also presented to test the strength and applicability of the results obtained.

New sets of orthogonal functions, which correspond to the first, second, third, and fourth kind Chebyshev polynomials with half-integer indexes, have been recently introduced. In this article, links of these new sets of irrational functions to the third and fourth kind Chebyshev polynomials are highlighted and their connections with the classical Chebyshev polynomials are shown.

In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas functions of fractional degree, and we show possible applications of these new functions. For the first kind, we compute the fractional Newton sum rules of any orthogonal polynomial set starting from the entries of the Jacobi matrix. For the second kind, the representation formulas for the fractional powers of a r×r matrix, already introduced by using the pseudo-Chebyshev functions, are extended to the Lucas case.

This paper provides novel generalizations by considering the generalized conformable fractional integrals for reverse Copson’s type inequalities on time scales. The main results will be proved using a general algebraic inequality, chain rule, Hölder’s inequality, and integration by parts on fractional time scales. Our investigations unify and extend some continuous inequalities and their corresponding discrete analogues. In addition, when α = 1, we obtain some well-known time scale inequalities due to Hardy, Copson, Bennett, and Leindler inequalities.

In this paper we discuss generalized two-variable Chebyshev polynomials and their relevant relations; in particular, by using their integral representations, we prove some operational identities. The approach is based on the generalized two-variable Hermite polynomials and the integral representations of ordinary Chebyshev polynomials of first and second kind. In addition, we discuss how the families of generalized Chebyshev polynomials can be used to prove some interesting properties related to ordinary Chebyshev polynomials of first and second kind. A fundamental role, as we see, is played by the powerful operational techniques verified by the families of generalized Hermite polynomials.

A plethora of higher order iterative methods, involving derivatives in algorithms, are available in the literature for finding multiple roots. Contrary to this fact, the higher order methods without derivatives in the iteration are difficult to construct, and hence, such methods are almost non-existent. This motivated us to explore a derivative-free iterative scheme with optimal fourth order convergence. The applicability of the new scheme is shown by testing on different functions, which illustrates the excellent convergence. Moreover, the comparison of the performance shows that the new technique is a good competitor to existing optimal fourth order Newton-like techniques.