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In this article, we consider the exact solutions to the simplified modified Camassa-Holm (SMCH) equation which has many potential applications in mathematical physics and engineering sciences. We examine the exact travelling wave solutions by means of the modified simple equation (MSE) method by making use of travelling transformation. The attained solutions are in the form of trigonometric and hyperbolic functions. We demonstrate that the method is more general, straightforward and powerful and can be used to examine more general travelling wave solutions of various kinds of fractional nonlinear differential equations arising in mathematical physics and better than other method. Finally, we show the graphical representation and discuss the physical significance of the obtained solutions for its definite values of the involved parameters through depicting 3D and 2D figures in order to know the physical phenomena.

Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differential equations (PDEs). In Caputo logic, the fractional-order derivative operator is measured. The Elzaki transform method and the residual power series method (RPSM) are combined in this novel technique. The suggested technique is based on a new version of Taylor’s series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates computing the fractional derivatives each time. As ERPSM just requires the concept of a zero limit, we simply need a few computations to get the coefficients. The novel technique solves nonlinear problems without the need for He’s and Adomian polynomials, which is an advantage over the other combined methods based on homotopy perturbation and Adomian decomposition methods. The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the presented method. Graphical significances are also identified for various values of fractional-order derivatives. As a result, the procedure is quick, precise, and easy to implement, and it yields outstanding results.

This study investigates the spectral and topological properties of rounded knot networks , a helical extension of phenylene quadrilateral structures, through signless Laplacian spectral analysis. Motivated by the need to understand how helical topology influences network dynamics and robustness, we derive exact analytical expressions for three key invariants: the Kirchhoff index , mean first‐passage time , and spanning tree count . By decomposing the signless Laplacian matrix into symmetric tridiagonal blocks Q + (Γ) and Q − (Γ), we obtain closed‐form eigenvalue solutions governed by helical periodicity following scaling. The results reveal that the Kirchhoff index grows approximately as O ( n 2 ), indicating enhanced global connectivity with network expansion, while the mean first‐passage time scales linearly with the network order, reflecting efficient diffusion behavior. The spanning tree enumeration shows exponential growth proportional to , demonstrating high structural redundancy and robustness. These findings establish precise mathematical links between helical topology, transport efficiency, and network reliability, offering potential implications for analyzing biopolymer helices and designing nanomaterials with tunable conductivity and resilience.

In quantitative structure-property and structure-activity relationships studies, several graph invariants, namely, topological indices have been defined and studied due to their numerous applications in computer networks, biotechnology, and nanochemistry. Topological indices are numeric parameters that describe the biological, physical, and chemical properties depending on the structure and topology of different chemical compounds. In this article, we inaugurated some degree-based novel indices, namely, geometric-harmonic GHI, harmonic-geometric HGI, neighborhood harmonic-geometric NHGI, and neighborhood geometric-harmonic NGHI indices and verified their chemical applicability. Furthermore, an attempt is made to calculate analytical closed formulas for different variants of silicon carbides and analyze the obtained results graphically.

This article deals with the upper bound of fourth-order Hankel and Toeplitz determinants for the convex functions which are defined by using the sine function. The main tools in this study are the coefficient inequalities for the class P of functions with positive real parts. Also, the investigation of the upper bound of the fourth-order Hankel determinant for 3-fold symmetric convex functions associated with the sine function is included.

In this manuscript, we present the generalized hypergeometric function of the type Fr r,r>2 and extension of the K Laguerre polynomial for the K extended Laguerre polynomials Ar,n,kαx. Additionally, we describe the K generating function, K recurrence relations, and KS Rodrigues formula.

The core objective of this article is to introduce and investigate a new class β−UCVqλA,B of convex functions associated with the conic domain defined by the Ruscheweyh q-differential operator. Many interesting properties such as sufficiency criteria, coefficient bounds, partial sums, and radius of convexity of order α for the functions of the said class are investigated here.

In this manuscript, we present the generalized hypergeometric function of the type F r   r , r > 2 and extension of the K Laguerre polynomial for the K extended Laguerre polynomials A r , n , k α x . Additionally, we describe the K generating function, K recurrence relations, and K S Rodrigues formula.

Background Data on emergency and critical care services in Somalia and other countries with fragile and conflict-affected (FCA) situations are limited, although the burden of emergency health conditions remain high. Improving emergency care services could significantly help improve health outcomes and realize the goals of achieving the Universal Health Coverage in these fragile countries. Methods We conducted a cross-sectional survey to assess the service availability and readiness for emergency and critical care (ECC) services in the country covering 131 hospitals in Somalia. The survey included both the public and private sector hospitals as well as those at the district, regional and national level hospitals representing the whole organizational structure of emergency health care system in the country. We administered the WHO Hospital Emergency Unit assessment tool which was slightly modified and adapted for this assessment. The survey included questions on the availability of staff, supplies/equipment, services, and systems. The respondents were hospital administrators and clinicians in outpatient departments, emergency departments and inpatient units. An overall median capacity score indicating the service availability and readiness for ECC was calculated for the country as well as for each hospital. Data were collected from December 2020 to March 2021. Result A total of 524 staff members across 131 hospitals participated in the survey. The median and interquartile range (IQR) ECC readiness score for all health facilities in the country was 0.31 (0.22–0.46) and only 26 (19.8%) facilities assessed had a median readiness score of more than 0.5 ( p value = 0.001). Using the cut-off point of 0.5, over 80% of the hospitals assessed were not considered ready to provide ECC services in the country. The third-level hospitals of the private sector including those facilities situated in predominantly urban areas were found to have better readiness to provide ECC services. User fees, lack of equipment and the absence of staff availability around the clock were identified as the most common barriers to emergency and critical care readiness. Conclusion This is the first study of its kind to be conducted in Somalia using a standardized tool and methodology and provides a comprehensive understanding of emergency and critical care services available in Somalia by facility type and levels of care. The study highlighted that significant capacity gaps exist at all levels in the provision of emergency care services especially in the public sector and at the first-level of care. Investment in emergency care services is urgently needed in the country bringing the primary care into the care continuum for ECC services along with implementing a set of cost-efficient interventions at the first-level of care given the country’s high burden of emergency health conditions. Clinical trial number Not applicable as this assessment was not a clinical trial.