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The present work introduces a novel approach, the Adomian Decomposition Formable Transform Method (ADFTM), and its application to solve the fractional order Sharma-Tasso-Olver problem. The method’s distinctive outcomes are highlighted through a comparative analysis with established non-local Caputo fractional derivatives and the non-singular Atangana–Baleanu (ABC) fractional derivatives. To provide a comprehensive understanding, the proposed ADFTM’s approximate solution is compared with the homotopy perturbation method (HPM) and residual power series method (RPSM). Further, numerical and graphical results demonstrate the reliability and accuracy of the ADFTM approach. The novel outcomes presented in this work emphasize its capability to address complex engineering problems effectively. By demonstrating its efficacy in solving the fractional order problems, the new ADFTM proves to be a valuable tool in solving scientific problems.

In this work, we mainly focus on the local existence and regularity of integral solutions for a class of nondensely defined partial neutral functional integrodifferential equations with unbounded delay. We use the theory of integrated resolvent operators introduced by Oka [H. Oka, Integrated resolvent operators, J. Integral Equations Appl. 7 1995, 2, 193–232]. Finally, we provide an example to demonstrate the basic findings of our work.

In this paper, we study the number of limit cycles bifurcated from the periodic orbits of a cubic uniform isochronous center with continuous and discontinuous quartic polynomial perturbations. Using the averaging theory of first order for continuous and discontinuous differential systems and comparing the obtained results, we show that the discontinuous systems have at least 6 more limit cycles than the continuous ones. This study needs some computations that have been verified using Maple.

In this article, we aim to define ℐ 2 ⁢ ( 𝒮 ) {\mathcal{I}_{2}(\mathcal{S})} -summability and ℐ 2 ⁢ ( 𝒮 θ p , r ) {\mathcal{I}_{2}(\mathcal{S}_{{\theta}_{p,r}})} -summability, and obtain interesting relationships among these by imposing certain conditions on p and r . Finally, we show that the space ℐ 2 ⁢ ( 𝒮 θ p , r ⁢ ( G , B , Y ) ) ∩ l ∞ 2 ⁢ ( V ) \mathcal{I}_{2}(\mathcal{S}_{{\theta}_{p,r}}(G,B,Y))\cap l^{2}_{\infty}(V) is a closed subspace of l ∞ 2 ⁢ ( V ) {l^{2}_{\infty}(V)} .

Unfortunately, eleven of our provinces were severely affected due to two severe earthquakes that occurred in our country, the Republic of Turkey, on February 6, 2023. As a result, thousands of buildings were destroyed and tens of thousands of our citizens lost their lives. From past to present, such disasters have occurred in many parts of our world and will continue to happen. In order to raise awareness for researchers and academicians reading our article, we will give a new definition of convexity in this article, and we will call it “earthquake convexity”. In this paper, we study some algebraic properties of the earthquake convexity. Then we compare the results obtained with both Hölder, Hölder–İşcan inequalities and power-mean, improved power-mean integral inequalities and show that the results obtained with Hölder–İşcan and improved power-mean inequalities are better than the others. Some applications to special means of real numbers are also given.

In this paper, by using the notions of λ-density, ( V , λ ) {(V,\lambda)} -summability and Orlicz function ϕ, we introduce a new concept of convergence, namely λ-statistically ϕ-convergence, as a generalization of the λ-statistical convergence and statistically ϕ-convergence. Based on this concept, we introduce a new sequence space S λ - ϕ {S_{\lambda}-\phi} and investigate some of its properties. Also, we find its relations with statistically ϕ-convergence, [ C , 1 ] ϕ {[C,1]_{\phi}} -summability and [ V , λ ] ϕ {[V,\lambda]_{\phi}} -summability. Finally, we introduce and investigate the concept of S λ - ϕ {S_{\lambda}-\phi} Cauchy sequences.

In this article, we discuss the relation theoretic aspects of multivalued weakly contractive mappings to prove fixed point results in the setting of metric spaces endowed with a certain binary relation. Our newly proved results generalize, extend, unify, enrich, sharpen and improve some well-known fixed point theorems of existing literature to the case of multivalued and contractive notion. We also incorporated an example and an application to find the solution of a Volterra-type integral inclusion.

In this paper, we prove the Titchmarsh theorem for Laguerre hypergroup K = [ 0 , + ∞ [ × R \mathbb{K}=[0,+\infty\mathclose{[}\times\mathbb{R} , via moduli of continuity of higher orders.

This paper proposes a new operational numerical method based on Touchard wavelets for solving fractional pantograph differential equations. First, we present an operational matrix of fractional integration as well as the fractional derivative of the Touchard wavelets. Then, by approximating the fractional derivative of the unknown function in terms of the Touchard wavelets and also by using collocation method, the original problem is reduced to a system of algebraic equations. Finally, to show the accuracy and the validity of the proposed technique, we provide some numerical examples.

We consider the l -th order linear fractional differential equations with constant coefficients. Here l ∈ ℕ {l\in\mathbb{N}} is the ceiling for the highest derivative of order α, l - 1 < α ≤ l {l-1<\alpha\leq l} . If β i < α {\beta_{i}<\alpha} are the other derivatives, the existing theory requires α - max ⁡ { β i } ≥ l - 1 {\alpha-\max\{\beta_{i}\}\geq l-1} for the existence of l linearly independent solutions. Thus, at most one derivative may have order greater than one, but all other derivatives must be between zero and one. We remove this essential restriction and construct l linearly independent solutions. With this aim, we remodel the series approaches and elaborate the multi-sum fractional series method in order to obtain the existence and linear independence results. We consider both Riemann–Liouville or Caputo fractional derivatives.

The role of fuzzy 𝛿-open set is highly significant in the study of fuzzy topology initiated by Ganguly and Saha [S. Ganguly and S. Saha, A note on 𝛿-continuity and 𝛿-connected sets in fuzzy set theory, Simon Stevin 62 (1988), 2, 127–141]. This article begins with the introduction of 𝛿-ℐ-open covers in a mixed fuzzy ideal topological space. After that, we introduce 𝛿-ℐ-compactness and then some properties of its are discussed therein. It is shown that the aforesaid compactness is the weaker form of fuzzy compactness. Moreover, we show that if we retopologize the fuzzy topology then in the new environment fuzzy 𝛿-ℐ-compactness and fuzzy compactness are equivalent. In addition, we introduce two different notions of continuity and investigate the behavior between fuzzy 𝛿-ℐ-compactness and fuzzy compactness.

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of the time fractional Black–Scholes equation derived by the fractional Brownian motion. Some exact solutions are obtained, the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, a new conservation theorem and a generalization of the Noether operators are developed to construct the conservation laws for the time fractional Black–Scholes equation.

In the current paper, we introduce the notion of statistical convergence of order α and strongly p -Cesàro summability of order α of sequences in the gradual normed linear spaces. We investigate several properties and a few inclusion relations of the newly introduced notions.

The objective of this study is to determine whether the existence of shared sets S containing both meromorphic (entire) functions and their higher derivatives, as well as powers of meromorphic functions and their differential polynomials, could result in uniqueness. The main focus is on determining the precise solutions to various differential equations. This problem is being studied in a broader context, specifically in set sharing.

The objective of this manuscript is to introduce the notion of multiplicative 𝔪 {\mathfrak{m}} -metric space, inspired by the concepts of multiplicative metric space and 𝔪 {\mathfrak{m}} -metric space. We propose a generalized notion of the distance function known as the multiplicative 𝔪 {\mathfrak{m}} -metric. Firstly, we present the fundamental definitions for the multiplicative 𝔪 {\mathfrak{m}} -metric space, then delve into the topological aspects, including the convergence of sequences and completeness. Additionally, we provide various illustrations to support our discussion. We also generalize some well-known contraction mappings and prove fixed point theorems on the complete multiplicative 𝔪 {\mathfrak{m}} -metric space. To complement our findings, we have included numerical results along with graphs to provide visual support for our conclusions. Furthermore, we explore the potential of utilizing the multiplicative 𝔪 {\mathfrak{m}} -metric to demonstrate the existence of the solution to a multiplicative integral equation.

In this paper, we introduce and investigate a subclass of analytic and bi-univalent functions which both f ⁢ ( z ) {f(z)} and f - 1 ⁢ ( z ) {f^{-1}(z)} are m -fold symmetric on the open unit disk. Furthermore, we find upper bounds for the initial coefficients | a m + 1 | {|a_{m+1}|} and | a 2 ⁢ m + 1 | {|a_{2m+1}|} for functions in this subclass. The results presented in this paper generalize and improve some recent works

This paper aims to study a generalized extended ( 2 + 1 ) {(2+1)} -dimensional Kadomstev–Petviashvili (KP) equation. The KP equation models several physical phenomena such as shallow water waves with weakly nonlinear restoring forces. We will use a variety of wave ansatz methods so as to extract bright, singular, shock waves also referred to as dark or topological or kink soliton solutions. In addition to soliton solutions, we will also derive periodic wave solutions and other analytical solutions based on the invariance surface condition. Moreover, we will establish the multiplier method to derive low-order conservation laws. In order to have a better understanding of the results, graphical structures of the derived solutions will be discussed in detail based on some selected appropriate parametric values in 2-dimensions, 3-dimensions and contour plots. The findings can well mimic complex waves and their underlying properties in fluids.