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Our manuscript is related to use Caputo fractional order derivative (CFOD) to investigate results of non-linear mode in plasma. We establish results for both temporal and spatial approximate solution. For the require results, we use reduction perturbation method (RPM) to find the analytical solution of the dust acoustic shock waves. Further, using the same technique we find the solitary wave potential and compared the solutions obtained with another very useful technique known as Homotopy perturbation method (HPM). The comparison of results for both approaches are more precise and agreed with the exact solution of the problem. Finally, we present graphical representation for different fractional order for both temporal and spatial approximate solution.

Abstract Let S n − 1 $\mathbb{S}^{n-1}$ denote unit sphere in R n $\mathbb{R}^{n}$ equipped with the normalized Lebesgue measure. Let Φ ∈ L s ( S n − 1 ) $\Phi \in L^{s}(\mathbb{S}^{n-1})$ be a homogeneous function of degree zero such that ∫ S n − 1 Φ ( y ′ ) d σ ( y ′ ) = 0 $\int _{\mathbb{S}^{n-1}}\Phi (y^{\prime})d \sigma (y^{\prime})=0$ , where y ′ = y / | y | $y^{\prime}=y/|y|$ for any y ≠ 0 $y\neq 0$ . The commutators of variable Marcinkiewicz fractional integral operator is defined as [ b , μ Φ ] β m ( f ) ( x ) = ( ∫ 0 ∞ | ∫ | x − y | ≤ s Φ ( x − y ) [ b ( x ) − b ( y ) ] m | x − y | n − 1 − β ( x ) f ( y ) d y | 2 d s s 3 ) 1 2 . $$ [b,\mu _{\Phi}]^{m}_{\beta }(f)(x )= \left ( \int \limits _{0} ^{ \infty }\left |\int \limits _{|x -y | \leq s} \frac{\Phi (x -y )[b(x )-b(y )]^{m}}{|x -y |^{n-1-\beta (x )}}f(y )dy \right |^{2} \frac{ds}{s^{3}}\right )^{\frac{1}{2}}. $$ In this paper, we obtain the boundedness of the commutators of the variable Marcinkiewicz fractional integral operator on grand variable Herz spaces K ˙ p ( ⋅ ) α ( ⋅ ) , q ) , θ ( R n ) ${\dot{K} ^{\alpha (\cdot ), q),\theta}_{ p(\cdot )}(\mathbb{R}^{n})}$ .

We study the comparative exact solutions of the Kairat-II equation describing the physical behaviors of nonlinear systems. The newly introduced Kairat equation has numerous applications in the fields of plasma physics, optical communications, differential geometry engineering, oceanography and physics. Two types of fractional operators known as β and M-truncated derivatives have been applied for creating the complex fractional Kairat-II equation. The two newly integrated methods, known as modified generalized Riccati equation mapping method and the generalized exponential rational function method approach, are under consideration to investigate the governing system. The implemented methodologies are distinguished by their efficacy, straightforwardness, and flexibility, which allow for the integration of diverse types of soliton solutions within a unified framework. In addition, to visualize the solution behaviors with different parameter values, we plot the different graphs with the associated parameter values under the effect of M-truncated and β-fractional derivatives.

In this article, we investigate the soliton wave dynamics of the fractional three-component coupled nonlinear Schrödinger equation. This equation is considered as a fundamental tool for describing the behavior of optical pulses in optical fibers. There is increasing interest in studying these equations as they can be used to explain a wide range of complex physical phenomena in various scientific and engineering fields, including nonlinear fiber optics, electromagnetic field waves, and signal processing through optical fibers. In order to obtain the required solutions, the first step is to apply the complex wave transformations with β-fractional derivative to obtain the nonlinear ordinary differential equations of the governing model. Furthermore, by applying the novel integration methods known as generalized Arnous method and multivariate generalized exponential rational integral function approach, different type of soliton solutions are extracted. Different sets and values of the physical parameters are used to study the optical soliton solutions of the studied system. Based on a comparative analysis of our results with existing techniques, this study concludes that our proposed solutions are novel. The results described in this work are able to enhance the nonlinear dynamical behavior of a given system and confirm the effectiveness of the methods used. Furthermore, the different graphs are sketched to visualize the solutions behavior with various parametric values. It is anticipated that obtained solutions will be significant in the study of wave propagation and other related fields.

Abstract This paper deals with the mathematical analysis of Tuberculosis by using fractal fractional operator. Mycobacterium TB is the bacteria that causes tuberculosis. This airborne illness mostly impacts the lungs but may extend to other body organs. When the infected individual coughs, sneezes or speaks, the bacterium gets released into the air and travels from one person to another. Five classes have been formulated to study the dynamics of this disease: susceptible class, infected of DS, infected of MDR, isolated class, and recovered class. To study the suggested fractal fractional model’s wellposedness associated with existence results, and boundedness of solutions. Further, the invariant region of the considered model, positive solutions, equilibrium point, and reproduction number. One would typically employ a fractional calculus approach to obtain numerical solutions for the fractional order Tuberculosis model using the Adams-Bashforth-Moulton method. The fractional order derivatives in the model can be approximated using appropriate numerical schemes designed for fractional order differential equations.

The main objective of the present paper is to establish the existence and uniqueness (EU) results for nonlinear fractional Langevin equation involving Liouville-Caputo generalized fractional derivative (GFD) of different order with non-local boundary conditions. The existence solution is obtained by using Krasnoselskii's fixed point theorem, and the uniqueness result is obtained by using the Banach contraction mapping principle. An example is introduced to validate the effectiveness of the results. The results are novel and provide an extension to some of the findings known in the literature.

This article secures the various wave structures of the fractional double dispersive equation, a significant nonlinear equation that describes the propagation of nonlinear waves within the elastic, uniform, and inhomogeneous Murnaghan’s rod. The model under discussion has a wide range of applications in science and engineering. Two recently developed analytical techniques known as the improved generalized Riccati equation mapping method and the multivariate generalized exponential rational integral function method have been applied to the proposed equation for the first time. A variety of solutions have been revealed such that dark, singular, bright-dark, bright, complex, and combined solitons. Furthermore, we include a diverse array of plots that illustrate the physical interpretation of the obtained solutions in relation to a number of significant parameters, thereby highlighting the impact of fractional derivatives. Within the context of the proposed model, these visualizations give a clear understanding of the behavior and characteristics of the solutions. This study’s results have the potential to enhance comprehension of the nonlinear dynamic characteristics exhibited by the specified system and validate the efficacy of the implemented techniques. The achieved results significantly enhance our understanding of nonlinear science and the nonlinear wave fields associated with more complex nonlinear models.

The magnitude and timing of extreme events such as debris and floodflows (collectively referred to as floodflows) in post-wildfire basins are difficult to measure and are even more difficult to predict. To address this challenge, a sensor ensemble consisting of noncontact, ground-based (near-field), Doppler velocity (velocity) and pulsed (stage or gage height) radars, rain gages, and a redundant radio communication network was leveraged to monitor flood wave velocities, to validate travel times, and to compliment observations from NEXRAD weather radar. The sensor ensemble (DEbris and Floodflow Early warNing System, DEFENS) was deployed in Waldo Canyon, Pike National Forest, Colorado, USA, which was burned entirely (100 percent burned) by the Waldo Canyon fire during the summer of 2012 (MTBS, 2020).Surface velocity, stage, and precipitation time series collected during the DEFENS deployment on 10 August 2015 were used to monitor and predict flood wave velocities and travel times as a function of stream discharge (discharge; streamflow). The 10 August 2015 event exhibited spatial and temporal variations in rainfall intensity and duration that resulted in a discharge equal to 5.01 cubic meters per second (m3/s). Discharge was estimated post-event using a slope-conveyance indirect discharge method and was verified using velocity radars and the probability concept algorithm. Mean flood wave velocities – represented by the kinematic celerity ck=2.619meterspersecond,m/s±0.556percent and dynamic celerity cd=3.533m/s±0.181percentandtheiruncertainties were computed. L-moments were computed to establish probability density functions (PDFs) and associated statistics for each of the at-a-section hydraulic parameters to serve as a workflow for implementing alert networks in hydrologically similar basins that lack data.Measured flood wave velocities and travel times agreed well with predicted values. Absolute percent differences between predicted and measured flood wave velocities ranged from 1.6 percent to 49 percent and varied with water slope, hydraulic radius, and depth. The kinematic celerity was a better predictor for steep slopes and wide flood plains associated with the Upper Waldo and Middle Waldo radar streamgages; whereas, the dynamic celerity was a better surrogate for shallow slopes and incised channels such as the Lower Waldo radar streamgage.The method demonstrates the potential extensibility of a post-wildfire warning system by (1) leveraging multiple systems (i.e., weather radar, near-field velocity and stage radars, and rain gages) for accurate and timely warnings of debris and floodflows, (2) establishing an order of operations to site, install, and operate near-field radars and conventional rain gages to record floodflows, forecast travel times, and document geomorphic change in this basin and hydrologically similar basins that lack data, and (3) communicating data operationally with the Colorado Department of Transportation engineering staff, National Weather Service forecasters, and emergency managers.

We take into account the bifurcation analysis of the generalized Pochhammer-Chree (PC) equation that describes the dynamics of several systems in science and engineering. The considered model is changed into a planar dynamical system by applying the Galilean transformation. The phase portraits are plotted by considering suitable values of the bifurcation parameters. The considered model is solved using the RK method to compute the supernonlinear and nonlinear wave solutions. All phase portraits and wave solutions are depicted in the phase plane by simply fixing the relevant parameters values. The equilibrium points are obtained, and the same are classified. Moreover, sensitive analysis for different initial value problems is applied to analyze the quasiperiodic, chaotic behavior and time series after introducing an extrinsic periodic perturbation term. In addition, the Lyapunov characteristic exponents, Poincare section and bifurcation diagrams are also discussed to examine the chaotic pattern of the model. Numerical simulation results show that changing the frequencies and amplitude values impacts the dynamical features of the considered model.

This research aims to investigate the approximate controllability of multivalued impulsive stochastic fractional differential inclusions in Hilbert space with ABC fractional-order derivatives. First, we confirm the existence of mild solutions for the proposed control system using stochastic analysis, resolvent operator theory, and the fixed point technique. Secondly, we discuss a new set of sufficient conditions for the approximate controllability of the systems. The results are obtained under the assumption that the associated linear system is approximately controllable. Finally, an example is provided to illustrate the obtained results.

Objective: To evaluate paroxysmal atrial fibrillation (AF) prevalence in Indian adults who completed 24-Hour Holter monitoring. Methods: A total of 23,847 patients (36.9 % women) were analyzed for AF duration using a software algorithm. Results: AF was diagnosed in 4153 (17.4 %) patients with a median AF duration of 13 min and 55 s. Conclusion: AF prevalence was high and largely untreated. The short duration of AF episodes indicates a low likelihood of detection during clinical visits, highlighting its potential underestimation in Indian healthcare.

Salt is a potent abiotic stress that arrests plant growth by impairing their physio-biochemical and molecular processes. However, it is unknown how the ABA signaling system and vacuolar-type Na+/H+ antiporter proteins induce stress tolerance in wheat (Triticum aestivum L.) seedlings. The present study aimed to identify salt-responsive proteins and signaling pathways involved in the resistance of wheat to salt stress. We explored the proteome profile, 20 amino acids, 14 carbohydrates, 8 major phytohormones, ion content, and salt tolerance genes in wheat (Triticum aestivum L., cv.) under 200 mM NaCl with control plants for six days. The results showed that amino acids such as alanine, serine, proline, glutamine, and aspartic acid were highly expressed under salt stress compared with control plants, suggesting that amino acids are the main players in salinity tolerance. The ABA signaling system was activated in response to salinity stress through the modulation of protein phosphatase 2C (PP2C) and ABA-responsive element binding factor (ABF), resulting in an ABA-mediated downstream response. Additionally, the vacuolar-type Na+/H+ antiporter was identified as a key protein in salt stress tolerance via compartmentalizing Na+ in the vacuole. Furthermore, a significant increase in the abundance of the 14-3-3 protein was noticed in salt-fed plants, suggesting that this protein plays an important role in Na+ compartmentalization. Moreover, up-regulation of ascorbate peroxidase (APX), glutathione-S-transferase (GST), and thioredoxin-scavenged reactive oxygen species resulted in improved plant growth under salt stress. These data will help to identify salt-responsive proteins that can be used in future breeding programs to develop salt-tolerant varieties.

Long-term overfertilization not only aggravates environmental costs, but reduces the quality and yield of fruit in dryland apple orchard, and this issue is closely associated with post-germination fertilization management. On the basis of multi-year observations, we developed a targeted fertigation strategy of FORD, i.e. formula fertilization (rational fertilizer scheme), on-demand fertilization in the sensitive period, reduced fertilization and delayed fertilization. Two-year field investigations were conducted with conventional furrow application (CK), integrated furrow and injection application (FI) and the optimized injection application under the FORD principle in a semiarid apple orchard. The data indicated that FI and FORD had a similar fruit yield of 57.2 Mg ha-1, significantly greater (by 4.5%) than CK. Yet, fruit quality (nutrition, flavor and appearance) was substantially improved under FORD, relative to FI and CK (P

Cubic Fermatean fuzzy sets offer a flexible and powerful tool for decision-making, permitting decision-makers to incorporate uncertainty and reluctance into their decision-making process, and enabling them to make more informed and effective decisions in complex and uncertain situations. We present cubic fermatean fuzzy set and operational laws. We offer some cubic fermatean fuzzy numbers, as well as some cubic fermatean fuzzy aggregation operators, such as the cubic fermatean Einstein fuzzy weighted geometric operator, the cubic fermatean Einstein fuzzy ordered weighted geometric operator, and the cubic fermatean Einstein fuzzy hybrid weighted geometric operator. To demonstrate the solution to complicated real-life problems, an algorithm for the MADM problem is established under a system of cubic fermatean fuzzy information. To check the possibility of proposed aggregation approaches, we provided a numerical example to choose the desirable best options by using conceived approaches. To disclose the suppleness and applicability of designed methods, sensitive study, and comparative study are illustrated by complementary findings of existing methods with presently developed approaches.

Employing an improved Leslie-Gower model, the Allee impact on the predator population and the fear influence on the prey population, a mathematical model of the interaction involving two populations, prey and predator, has been explored in the current work. The influence of the memory effect on the evolving behaviour of the model is integrated using a well-known fractional operator known as the Caputo fractional derivative of order (0, 1]. We were inspired to do this study because further research is desired to fully comprehend the dynamics of the Allee impact, fear, and Caputo fractional order. The innovative aspect of the current work is the consideration of the Caputo fractional derivative, which enhances the consistency domain of the system. To support the model's biological resilience and accuracy, the existence, uniqueness, positivity, and boundedness of the solution are provided. On the origin, axial, and interior, four distinct types of equilibrium points are distinguished along with the prerequisites for their existence. We additionally stated about the Hopf bifurcation of our proposed model at the interior equilibrium point in terms of fractional order and the fear influence as the bifurcation factors. To illustrate how various biological characteristics affect the dynamics of the solutions, numerical simulations are offered.

The study of soliton theory plays a crucial role in the telecommunication industry’s utilization of nonlinear optics. The principal area of research in the field of optical solitons revolves around optical fibers, metamaterials, metasurfaces, magneto-optic waveguides, and other related technologies. Therefore, the examination of these solitons received significant attention from scholars in recent years. Optical solitons refer to electromagnetic waves that are confined within nonlinear dispersive medium, wherein the balance between dispersion and nonlinearity effects enables the intensity to remain constant. In this manuscript, the dynamical behavior of Kudryashov’s equation is discussed with the assistance of truncated M-fractional derivative. Kudryashov’s equation is a mathematical model utilized to characterize complex phenomena in telecommunications and transmission technology. It describes the propagation of nonlinear pulses in optical fibers. Different forms of soliton solutions like bright, dark, singular and combo solitions have been extracted. Moreover, hyperbolic, periodic and Jacobi elliptic function solutions are recovered. Two recently modern integration tools like Φ6-expansion method and modified generalized exponential rational function method have been adopted to recover the solutions. The used methods not only provides previously extracted solutions but also secures new solutions. In order to visually depict the output, a variety of graphs featuring distinct shapes are produced in response to appropriate parameter values. The results obtained from this research indicate that the chosen methodologies are effective in improving the understanding of nonlinear dynamical phenomena. A large number of engineers who employ engineering models are expected to find this research interesting. The results demonstrate that the selected methods are practical, straightforward to implement, and applicable to intricate systems across numerous domains, with a specific emphasis on the field of optical fibers. The findings indicate that the system may contain a considerable number of soliton structures.

Nowadays, the study of fractional differential equations (FDEs) is one of the most interesting research topics. In this article, we study a novel nonlinear fractional order Langevin system involving delays. The primary focus of this article is to analyze the relative controllability of a nonlinear integrodifferential fractional Langevin dynamical system with multiple delays in control for finite-dimensional spaces. Fundamentals of the fixed-point theory are used to achieve the desired results. Using Schauder’s fixed theorem, sufficient conditions for the controllability are established and the Mittag-Leffler matrix function defines the controllability Grammian matrix. Additionally, the derived results are validated through an application.

In contrast to fossil fuels, the sun has more than enough energy to supply the entire world's energy needs. The sole constraint on solar energy as a renewable resource is our capacity to efficiently and economically convert it to electricity. In this work, we take advantage of the fractional derivative, we introduce the dynamics of the solar heating model. We incorporated the Atangana-Baleanu derivative (ABC) in our analysis. Then the solutions of our fractional system are investigated for existence and uniqueness. In order to visualize the fractional order model solution, we used a novel and trendy numerical method to represent the dynamics of different parameters of the nonlinear ordinary differential equation system. It is shown that the proposed ODE models are valid and efficient for fractional order data for a functioning solar heating system.

We are concerned with the analysis of the neural networks of worms in wireless sensor networks (WSN). The concerned process is considered in the form of a mathematical system in the context of fractal fractional differential operators. In addition, the Banach contraction technique is utilized to achieve the existence and unique outcomes of the given model. Further, the stability of the proposed model is analyzed through functional analysis and the Ulam-Hyers (UH) stability technique. In the last, a numerical scheme is established to check the dynamical behavior of the fractional fractal order WSN model.

This research paper focuses on establishing a framework for grand Herz-Morrey spaces defined over the $ p $-adic numbers and their associated $ p $-adic intrinsic square function. We will define the ideas of grand $ p $-adic Herz-Morrey spaces with variable exponent $ {M\dot{K} ^{\alpha, u), \theta}_{ s(\cdot)}(\mathbb{Q}^n_p)} $ and $ p $-adic intrinsic square function. Moreover, the corresponding operator norms are estimated. Grand $ p $-adic Herz-Morrey spaces with variable exponent is the generalization of $ p $-adic Herz spaces. Our main goal is to obtain the boundedeness of $ p $-adic intrinsic square function in grand $ p $-adic Herz-Morrey spaces with variable exponent $ {M\dot{K} ^{\alpha, u), \theta}_{ s(\cdot)}(\mathbb{Q}^n_p)} $. The boundedness is proven by exploiting the properties of variable exponents in these function spaces.

The fractional Hardy-type operators of variable order is shown to be bounded from the grand Herz spaces $ {\dot{K} ^{a(\cdot), u), \theta}_{ p(\cdot)}(\mathbb{R}^n)} $ with variable exponent into the weighted space $ {\dot{K} ^{a(\cdot), u), \theta}_{\rho, q(\cdot)}(\mathbb{R}^n)} $, where $ \rho = (1+|z_1|)^{-\lambda} $ and $ {1 \over q(z)} = {1 \over p(z)}-{\zeta (z) \over n} $ when $ p(z) $ is not necessarily constant at infinity.

Let $ \mathbb{S}^{n-1} $ denotes the unit sphere in $ \mathbb{R}^n $ equipped with the normalized Lebesgue measure. Let $ \Phi \in L^r(\mathbb{S}^{n-1}) $ be a homogeneous function of degree zero. The variable Marcinkiewicz fractional integral operator is defined as $ \mu _{\Phi} (f)(z_1) = \left( \int \limits _0 ^ \infty \left|\int \limits _{|z_1-z_2| \leq s} \frac{\Phi(z_1-z_2)}{|z_1-z_2|^{n-1-\zeta(z_1)}}f(z_2)dz_2\right|^2 \frac{ds}{s^3}\right)^{\frac{1}{2}}. $ The Marcinkiewicz fractional operator of variable order $ \zeta(z_1) $ is shown to be bounded from the grand Herz-Morrey spaces $ {M\dot{K} ^{\alpha(\cdot), u), \theta}_{\beta, p(\cdot)}(\mathbb{R}^n)} $ with variable exponent into the weighted space $ {M\dot{K} ^{\alpha(\cdot), u), \theta}_{\beta, \rho, q(\cdot)}(\mathbb{R}^n)} $ where $ \rho = (1+|z_1|)^{-\lambda} $ and $ {1 \over q(z_1)} = {1 \over p(z_1)}-{\zeta(z_1) \over n} $ when $ p(z_1) $ is not necessarily constant at infinity.

This article examines the third-order non-linear fractional Westervelt model by using a newly developed integration tool termed the new extended direct algebraic approach. The different wave structures to the considered model are secured in various forms, including bright, dark, and combo solitons. The application of wave structures is advantageous in the examination of sound wave propagation and high amplitude phenomena in the fields of medical imaging and therapy. These solutions are effective in facilitating ultrasound propagation in tissue, underwater acoustics, acoustic cavitation, acoustic levitation, and other related applications. The internal tissue of human beings can be visualized and studied through the use of ultrasound imaging technologies in the field of medicine. This technology possesses numerous uses in both industrial and medicinal sectors. There has been a growing interest in fractional nonlinear partial differential equations due to their ability to describe various complex occurrences and exhibit more dynamic architectures of localized wave solutions. On employing accurate parameter values, multiple graphical representations are generated to provide the visual depiction of the acquired results. The results of this study indicate that the chosen methodology effectively improves nonlinear dynamical processes. The findings indicate that the selected methodology demonstrates efficacy, feasibility, and versatility when applied to intricate systems across several domains, with a special emphasis on ultrasonic imaging. The findings indicate that the system exhibits a potentially significant abundance of soliton structures.

Summary: The 5′-deoxyadenosine deaminase (DADD), a member of the amidohydrolase family regulates biological purine metabolism. In this study, bioinformatic analysis, overexpression and knockdown of GhdadD gene were detected to identify its potential role in drought and salt stress tolerance. The results revealed that GhdadD was induced by ABA, Auxin, MBS and light responsive elements. In transgenic Arabidopsis, seed germination rate and root length were increased under drought or salt stress. GhdadD overexpressed seedlings resulted in higher plant height, less leaf damage and lower ion permeability. The expression of osmotic stress and ABA-responsive genes were up regulated. While in GhdadD-silenced cotton seedlings, CAT, SOD activity and soluble sugar content were reduced, MDA content was increased, and the stoma opening was depressed under drought or salt stress. Some osmics stress marker genes were also up regulated. These data indicating that GhdadD enhanced plant resistance to drought and salt stress through ABA pathways.

In the present study we have presented the notion of FUZZY BAYESIAN DECISION TECHNIQUE and combined the idea of the Fuzzy TOPSIS technique and entropy. We define the new ideas of fuzzy TOPSIS technique and entropy. So, we introduce the TOPSIS method and entropy, and the weights of the DMs are used. We proposed an MCDM technique based on TOPSIS and entropy. We focus on parameter different solutions of Fuzzy TOPSIS Positive ideal and Negative ideal solutions efficient decision making. Also, we provide a numerical example to elucidate the proposed technique stage by stage. Lastly, we compare the explanations of the current problem with the many existing MCGDM approaches to deliver the skills and rationality of the offered technique. We also provide a sensitivity study by shifting the entropy to establish the weights of the criteria underneath the dominant entropy measure meaning.

The goal of this article is to study the existence and unique solutions of coupled pantograph discrete fractional order difference equations. The Banach contraction principal method and fixed point theorems are used to prove the existence and uniqueness criterion of the suggested coupled system associated with a discrete fractional operator in the Caputo sense. In addition, stability analysis for the considered discrete pantograph fractional order coupled system is investigated by Hyers–Ulam theoretical approach. Finally, as an application, an example is presented to illustrate the obtained results.

Abstract This paper aims to extend the Caputo–Atangana–Baleanu ( A B C $ABC$ ) and Riemann–Atangana–Baleanu ( A B R $ABR$ ) fractional derivatives with respect to another function, from fractional order μ ∈ ( 0 , 1 ] $\mu \in (0,1]$ to an arbitrary order μ ∈ ( n , n + 1 ] $\mu \in (n,n+1]$ , n = 0 , 1 , 2 , … $n=0,1,2,\dots $ . Also, their corresponding Atangana–Baleanu (AB) fractional integral is extended. Additionally, several properties of such definitions are proved. Moreover, the generalization of Gronwall’s inequality in the framework of the AB fractional integral with respect to another function is introduced. Furthermore, Picard’s iterative method is employed to discuss the existence and uniqueness of the solution for a higher-order initial fractional differential equation involving an A B C $ABC$ operator with respect to another function. Finally, examples are given to illustrate the effectiveness of the main findings. The idea of this work may attract many researchers in the future to study some inequalities and fractional differential equations that are related to AB fractional calculus with respect to another function.

This article aims to demonstrate the formation of entropy due to variable thermal conductivity, radiation, and fluid friction irreversibilities for a three-dimensional upper-convected Maxwell (UCM) fluid. The fluid motion occurs as a result of exponential stretching sheets. Separate discussions are held regarding the entropy generation related to the prescribed surface temperature and prescribed surface heat flux. Additionally, the heat transport mechanism is examined in the presence of thermal radiation. The governing physical situation is first modeled and then solved by using the homotopy analysis method to acquire the solution. The physical importance of relevant flow parameters is shown graphically and in tabular form. It is noted that the entropy generated is reduced with an increase in the thermal radiation parameter. Streamline patterns are also drawn for two- and three-dimensional UCM fluid models. Finally, the current analytical solution is found to be in agreement with the solutions in the literature.

This study is confined to the numerical evaluation of variable density and magnetohydrodynamics influence on Williamson Sakiadis flow in a porous space. In this study, Joule heating, dissipation, heat generation effect on optically dense gray fluid is encountered. The inclined moving surface as flow geometry is considered to induce the fluid flow. A proposed phenomenon is given a mathematical structure in partial differential equations form. These partial differential equations are then made dimensionless using dimensionless variables. The obtained dimensionless model in partial differential equations is then changed to ordinary differential equations via stream function formulation. A set of transformed equations has been solved with bvp4c solver. The numerical fallout of velocity field, temperature field, skin friction, and heat transfer rate are illustrated in graphs and tables with flow parametric variations. Conclusion is drawn that mounting values of density variation parameter confirm the reduction in velocity field and augmentation in temperature of the fluid. When Williamson fluid parameter enhances, both fluid velocity and temperature are rising correspondingly. Growing magnitudes of the magnetic number, radiation parameter, heat generation, and Eckert number rise the temperature of the fluid. A rise in a porous medium parameter weakens the fluid velocity. Skin friction is reducing as radiation parameter and density variation parameter are increased.The present solutions are compared to those that have already been published in order to validate the current model. The comparison leads to the conclusion that the two outcomes are in excellent agreement, endorsing the veracity of the current answers.

The primary focus of this article is to provide sufficient conditions for the Ulam–Hyers stability of neutral delay fractional differential equations involving Hilfer fractional derivatives and Erdelyi–Kober fractional integral boundary conditions. The fixed point approach is utilized to prove the existence and uniqueness of mild solutions for the proposed problem. In the end, the derived results are validated through an illustrative example.

In this manuscript, the main objective is to analyze the existence, uniqueness, (EU) and stability of positive solution for a general class of non-linear fractional differential equation (FDE) with fractional differential and fractional integral boundary conditions utilizing $ \phi_p $-Laplacian operator. To continue, we will apply Green's function to determine the suggested FDE's equivalent integral form. The Guo-Krasnosel'skii fixed point theorem and the properties of the $ p $-Laplacian operator are utilized to derive the existence results. Hyers-Ulam (HU) stability is additionally evaluated. Further, an application is presented to validate the effectiveness of the result.

In this article, we investigate the existence of unique maximal and minimal solutions for a coupled differential system in terms of generalized fractional derivative with arbitrary order. The iterative technique of a fixed point operator together with the properties of green's function are used basically. Moreover, we investigate the generalized Ulam-Hyers stability of the solution for the given coupled system. Finally, some examples are given to illustrate the theoretic results.

Herein the research objects, a hyperbolic quasi-linear system of governing equations was solved by an asymptotic method (far-field technique) with explaining a 1-D unsteady planar and cylindrically symmetric flows in magnetogasdynamics. The evolution equation was obtained by generalized Burger's equation. A relatively accurate systematic result of the evolution equation was gotten by us through the analytic homotopy analysis method (HAM). We are allowed by the method to determine the various effects of nonlinearity and geometrical spreading. One of the fundamental problems of conservation laws are represented by the non-linear waves from preliminary data.

Taking into account the significance of spherical hesitant fuzzy sets, this research concentrates on an innovative multi-criteria group decision-making technique for dealing with spherical hesitant fuzzy (SHF) situations. To serve this purpose, we explore SHF Aczel Alsina operational laws such as the Aczel-Alsina sum, Aczel-Alsina product and Aczel-Alsina scalar multiplication as well as their desirable characteristics. This work is based on the fact that aggregation operators have significant operative adaptability to aggregate the uncertain information under the SHF context. With the aid of Aczel-Alsina operators, we develop SHF Aczel-Alsina geometric aggregation operators to address the complex hesitant uncertain information. In addition, we describe and verify several essential results of the newly invented aggregation operators. Furthermore, a decision aid methodology based on the proposed operators is developed using SHF information. The applicability and viability of the proposed methodology is demonstrated by using a case study related to breast cancer treatment. Comprehensive parameter analysis and a systematic comparative study are also carried out to ensure the dependability and validity of the works under consideration.

The subject of this work is the existence and Mittag-Leffler-Ulam (MLU) stability of solutions for fractional pantograph equations with three sequential fractional derivatives. Sufficient conditions for the existence and uniqueness of solutions are constructed by utilizing well-known classical fixed point theorems such as the Banach contraction principle, and Leray-Schauder nonlinear alternative. The generalized singular Gronwall's inequality is used to show the MLU stability results. An illustrated example is provided to support the main findings.

Many studies have shown that faced with an epidemic, the effect of fear on human behavior can reduce the number of new cases. In this work, we consider an SIS-B compartmental model with fear and treatment effects considering that the disease is transmitted from an infected person to a susceptible person. After model formulation and proving some basic results as positiveness and boundedness, we compute the basic reproduction number $ \mathcal R_0 $ and compute the equilibrium points of the model. We prove the local stability of the disease-free equilibrium when $ \mathcal R_0 < 1 $. We study then the condition of occurrence of the backward bifurcation phenomenon when $ \mathcal R_0\leq1 $. After that, we prove that, if the saturation parameter which measures the effect of the delay in treatment for the infected individuals is equal to zero, then the backward bifurcation disappears and the disease-free equilibrium is globally asymptotically stable. We then prove, using the geometric approach, that the unique endemic equilibrium is globally asymptotically stable whenever the $ \mathcal R_0 > 1 $. We finally perform several numerical simulations to validate our analytical results.

In this paper, the use of the Fermatean fuzzy number (FFN) in a significant research problem of disaster decision-making by defining operational laws and score function is demonstrated. Generally, decision control authorities need to brand suitable and sensible disaster decisions in the direct conceivable period as unfitting decisions may consequence in enormous financial dead and thoughtful communal costs. To certify that a disaster comeback can be made, professionally, we propose a new disaster decision-making (DDM) technique by the Fermatean fuzzy Schweizer-Sklar environment. First, the Fermatean fuzzy Schweizer-Sklar operators are employed by decision-makers to rapidly analyze their indefinite and vague assessment information on disaster choices. Then, the DDM technique based on the FFN is planned to identify highly devastating disaster choices and the best available choices. Finally, the proposed regret philosophy DDM technique is shown functional to choose the ideal retort explanation for a communal fitness disaster in Pakistan. The dominance and realism of the intended technique are further defensible through a relative study with additional DDM systems.

Abstract Background Current climate change scenarios are posing greater threats to the growth and development of plants. Thus, significant efforts are required that can mitigate the negative effects of drought on the cotton plant. GDSL esterase/lipases can offer an imperative role in plant development and stress tolerance. However, thesystematic and functional roles of the GDSL gene family, particularly in cotton under water deficit conditions have not yet been explored. Results In this study, 103, 103, 99, 198, 203, 239, 249, and 215 GDSL proteins were identified in eight cotton genomes i.e., Gossypium herbaceum (A1), Gossypium arboretum (A2), Gossypium raimondii (D5), Gossypium hirsutum (AD1), Gossypium barbadense (AD2), Gossypium tomentosum (AD3), Gossypium mustelinum (AD4), Gossypium darwinii (AD5), respectively. A total of 198 GDSL genes of Gossypium hirsutum were divided into eleven clades using phylogenetic analysis, and the number of GhirGDSL varied among different clades. The cis-elements analysis showed that GhirGDSL gene expression was mainly related to light, plant hormones, and variable tense environments. Combining the results of transcriptome and RT-qPCR, GhirGDSL26 (Gh_A01G1774), a highly up-regulated gene, was selected for further elucidating its tole in drought stress tolerance via estimating physiological and biochemical parameters. Heterologous expression of the GhirGDSL26 gene in Arabidopsis thaliana resulted in a higher germination and survival rates, longer root lengths, lower ion leakage and induced stress-responsive genes expression under drought stress. This further highlighted that overexpressed plants had a better drought tolerance as compared to the wildtype plants. Moreover, 3, 3’-diaminobenzidine (DAB) and Trypan staining results indicated reduced oxidative damage, less cell membrane damage, and lower ion leakage in overexpressed plants as compared to wild type. Silencing of GhirGDSL26 in cotton via VIGS resulting in a susceptible phenotype, higher MDA and H2O2 contents, lower SOD activity, and proline content. Conclusion Our results demonstrated that GhirGDSL26 plays a critical role in cotton drought stress tolerance. Current findings enrich our knowledge of GDSL genes in cotton and provide theoretical guidance and excellent gene resources for improving drought tolerance in cotton.

We consider saddle point problem and proposed an updating $ QR $ factorization technique for its solution. In this approach, instead of working with large system which may have a number of complexities such as memory consumption and storage requirements, we computed $ QR $ factorization of matrix $ A $ and then updated its upper triangular factor $ R $ by appending the matrices $ B $ and $ \left(\begin{array}{cc} B^T & -C \\ \end{array} \right) $ to obtain the solution. The $ QR $ factorization updated process consisting of updating of the upper triangular factor $ R $ and avoid the involvement of orthogonal factor $ Q $ due to its expensive storage requirements. This technique is also suited as an updating strategy when $ QR $ factorization of matrix $ A $ is available and it is required that matrices of similar nature be added to its right end or at bottom position for solving the modified problems. Numerical tests are carried out to demonstrate the applications and accuracy of the proposed approach.

The purpose of this article is to study the convergence, stability and data dependence results of an iterative method for contractive-like mappings. The concept of stability considered in this study is known as $ w^2 $-stability, which is larger than the simple notion of stability considered by several prominent authors. Some illustrative examples on $ w^2 $-stability of the iterative method have been presented for different choices of parameters and initial guesses. As an application of our results, we establish the existence, uniqueness and approximation results for solutions of a nonlinear Volterra integral equation with delay. Finally, we provide an illustrative example to support the application of our results. The novel results of this article extend and generalize several well known results in existing literature.

This manuscript is devoted to establishing some theoretical and numerical results for a nonlinear dynamical system under Caputo fractional order derivative. Further, the said system addresses an infectious disease like COVID-19. The proposed system involves natural death rates of susceptible, infected and recovered classes respectively. By using nonlinear analysis feasible region and boundedness have been established first in this study. Global and Local stability analysis along with basic reproduction number have also addressed by using the next generation matrix method. Upon using the fixed point approach, existence and uniqueness of the approximate solution for the mentioned problem has also investigated. Some stability results of Hyers-Ulam (H-U) type have also discussed. Further for numerical treatment, we have exercised two numerical schemes including modified Euler method (MEM) and nonstandard finite difference (NSFD) method. Further the two numerical schemes have also compared with respect to CPU time. Graphical presentations have been displayed corresponding to different fractional order by using some real data.

IntroductionLow agricultural nutrient input efficiency remains a significant impediment for crop production globally. To address this issue in cotton agroecosystems, there is a need to develop sustainable crop nutrient management strategies to achieve high crop yields. We hypothesized that organic liquid fertilizer (OF) combined with reduced chemical fertilizer (CF) would enhance cotton yield by improving leaf functioning and soil properties. However, the underlying mechanism and its related process is poorly understood.MethodsThis study explored the effects of OF combined with reduced CF on cotton yield, physiology and soil properties. Treatments included a single application of CF (CF: N, P2O5 and K2O applied at 228, 131 and 95 kg ha−1) and combined applications of OF and CF (OF0.6−OF1.4) in the following ratios: OF0.6, OF+60% CF; OF0.8, OF+80% CF; OF1.0, OF+100% CF; OF1.2, OF+120% CF; OF1.4, OF+140% CF. Results and discussionThe result showed that compared with CF, OF0.8, OF1.0 and OF1.2 increased soil organic matter (SOM) content by 9.9%, 16.3% and 23.7%, respectively. Compared with CF, the OF0.6, OF0.8, OF1.0, and OF1.2 treatments increased leaf area (LA) by 10.6−26.1%, chlorophyll content (Chl content) by 6.8−39.6%, and the efficiency of photosystem II (PSII) light energy (Y(II)), electron transfer rate of PSII (ETR) and photochemical quenching (qP) by 3.6−26.3%, 4.7−15.3% and 4.3−9.8%, respectively. The OF0.8 treatment increased net photosynthetic rate (Pn), stomatal conductance (Gs) and transpiration rate (E) by 22.0%, 27.4% and 26.8%, respectively, resulting in higher seed cotton yield. The seed cotton yield and economic coefficient were positively correlated with Pn, E, Gs and Y(II) from the full boll stage to the boll opening stage. In summary, the OF0.8 treatment can maintain a high SOM content and photosynthetic performance with reduced chemical fertilizer input without sacrificing yield. The integration of OF+80% CF (OF0.8) is a promising nutrient management strategy for highly efficient cotton production under mulch drip irrigation systems.

The Era of data is transubstantiating into a Big Data model in this technological world in the early 21st century. In 2005, Roger Mougalas coined a combination of data for this future world of the human race. The information helps to find specific solutions for any physical problem under Catastrophic circumstances in high populations such as Covid-19. To store massive data and historical events in a computer, the possibility of damage occurred to the complete data. Hence, viruses are a crucial threat to such data worth millions and billions. For this purpose, we spend enormous costs and efforts to build defensive strategies to save that information. Analyzing the expansion and extension of viruses helps to protect data and prevent viruses. In this manuscript, we study optimal control analysis for the suggested model in the sense of the Atangana–Baleanu derivative (AB-derivative). We employed a fixed point theorem to analyze the solutions for the fractional order computer virus model. We verified the results numerically and expressed them graphically.

Introduction: Biochar and biofertilizer have emerged as sustainable soil amendments to improve soil fertility and quality. However, it is necessary to determine how biochar’s various particle sizes and biofertilizers affect soil microbial biomass carbon (MBC), microbial biomass nitrogen (MBN), and CO2 sequestration.Methods: This research examined two rhizobia inoculum products—Biozote-N and Rhizogold—along with three biochar particle sizes (

Abstract Background Advanced understanding of tumor biology has recently revealed the complexity of cancer genetics, intra/inter-tumor heterogeneity, and diverse mechanisms of resistance to cancer treatment. In turn, there has been a growing interest in cancer prevention and minimizing exposure to potential environmental carcinogens that surround us. In the 1980s, several chemical carcinogens, including perchloroethylene (PCE), trichloroethylene (TCE), and benzene, were detected in water systems supplying Camp Lejeune, a US Marine Corps Base Camp located in North Carolina. Case presentation This article presents three cases of cancer patients who have lived at Camp Lejeune, and, decades later, came to our clinic located 1000 miles from the original exposure site. The first patient is a young Caucasian man who was diagnosed with T cell acute lymphoblastic leukemia at the age of 37, and the second patient is a Caucasian man who had multiple types of cancer in the prostate, lung, and colon as well as chronic lymphocytic leukemia in his 60s and 70s. The third patient is another Caucasian man who had recurrent skin cancers of different histology, namely basal cell carcinomas, squamous cell carcinomas, and melanoma, from his 50s to 70s. Conclusions The US Congress passed the Honoring America’s Veterans and Caring for Camp Lejeune Families Act in 2012, which covers appropriate medical care for the people affected by the contamination. We hope that this article raises awareness about the history of Camp Lejeune’s water contamination among cancer care providers, so the affected patients can receive appropriate medical coverage and cancer screening across the country.

Summary: A common technique for uncovering intra-tumor genomic heterogeneity (ITH) is variant detection. However, it can be challenging to reliably characterize ITH given uneven sample quality (e.g., depth of coverage, tumor purity, and subclonality). We describe a protocol for calling point mutations and copy number alterations using sequencing of multiple related clinical patient samples across diverse tissue, optimizing for sensitivity with specificity. The ith.Variant pipeline can be run on single- or multi-region whole-genome and whole-exome sequencing.For complete details on the use and execution of this protocol, please refer to Sun et al. (2017).1 : Publisher’s note: Undertaking any experimental protocol requires adherence to local institutional guidelines for laboratory safety and ethics.

The concept of spherical hesitant fuzzy set is a mathematical tool that have the ability to easily handle imprecise and uncertain information. The method of aggregation plays a great role in decision-making problems, particularly when there are more conflicting criteria. The purpose of this article is to present novel operational laws based on the Yager t-norm and t-conorm under spherical hesitant fuzzy information. Furthermore, based on the Yager operational laws, we develop the list of Yager weighted averaging and Yager weighted geometric aggregation operators. The basic fundamental properties of the proposed operators are given in detail. We design an algorithm to address the uncertainty and ambiguity information in multi-criteria group decision making (MCGDM) problems. Finally, a numerical example related to Parkinson disease is presented for the proposed model. To show the supremacy of the proposed algorithms, a comparative analysis of the proposed techniques with some existing approaches and with validity test is presented.

With the global food deficit increasing and rising climate change issues, there is a need to find green solutions to improve soil fertility and productivity while enhancing soil biochemical quality and reducing the ecological impact of agriculture. Biochar is a potentially cost-effective, carbonaceous resource with many agricultural and environmental applications. As a soil amendment, it improves soil physical and biochemical properties and increases soil fertility and productivity—particularly over the long-term—increasing soil aggregation, water retention, pH, and microbial activities, thus, improving overall soil quality, potentially helping to reduce chemical fertilizer needs over time. The extent of biochar’s impact on soil physiochemical properties varies depending on biochar source, type, size, inherent soil characteristics, cropping system, etc. Moreover, biochar has significant potential in soil and water remediation, especially through its unique adsorption and chemical properties capable to capture and immobilize pollutants such as metal(loid)s, organic pollutants, and hazardous emerging contaminants such as microplastics. Further, biochar has also emerged as a key strategic, cost-effective material to tackle global issues such as climate change mitigation, reducing the net greenhouse gas emission to minimize global warming potential. However, a knowledge gap remains as to understanding the long-term persistence of biochar on agroecosystem, optimal biochar application rate for the diversity of biochar-soil-crop-environmental conditions, interaction of biochar with inherent soil carbon stock, specific mechanisms of biochar’s effect on soil biotic properties, quantification of carbon sequestration, greenhouse gas emissions, synergy or potential antagonistic effects with other carbon sources such as compost, manure, residues, etc., its modification for environmental applications and associated environmental and human risks over long-term. Further research is needed to evaluate the long-term impacts of types and sizes of biochar on overall soil quality to recommend suitable application practices based on soil management and cropping system. Also, its environmental applications need to be finetuned for wider and target specific applications to tackle pressing environmental issues such as soil and water pollution.