In this study, we develop a novel stage-structured fractional-order prey-predator model that captures more realistic ecological dynamics by incorporating the effects of juvenile hunting behavior-particularly those arising from structural habitat complexity and shoot bombardment. These factors influence the viability and survival of juvenile predators, leading to potential deviations in population predictions. To account for the inherent memory and hereditary properties in ecological interactions, we employ fractional-order derivatives, thereby extending the classical prey-predator framework. The proposed model accounts for maturation delays, stage-specific interactions, and differential survival rates, offering a more comprehensive understanding of population stabilization across multiple life stages. Our analysis highlights how structural complexity in the habitat critically affects juvenile hunting efficiency and the associated maladroit costs, such as increased mortality due to unsuccessful hunting attempts during key developmental periods. Both analytical and numerical techniques are employed to investigate the stability and bifurcation behavior of the system under varying ecological parameters. The results underscore the importance of habitat features and predator limitations in shaping population dynamics and ecological resilience. These insights contribute to the broader understanding of biodiversity conservation and the strategic management of ecosystems under natural and anthropogenic pressures.

The Hilfer-Prabhakar (HP) derivatives are advance class of fractional operators with nonlocal features, memory effects, and nonexponential decay. These derivatives are generally used in study of heterogeneous systems, anomalous diffusion, dielectric spectroscopy, free electron lasers, fractional heat equations, and Cauchy problems. In this study, we derive the Kharrat-Toma (KT) transforms of the kernel function of the k-Prabhakar integral (k-PI), k-Prabhakar fractional derivative, its regularized form, and the k-Hilfer-Prabhakar fractional derivative (k-HPFD). We also establish the relationship between the k-Prabhakar fractional derivative and its regularized form for an absolutely continuous function using KT transform operations. Similarly, the KT transforms of k -HPFD and its regularized variant have been computed. Moreover, we establish the relationship between the k-HPFD and its regularized form using KT transform operations. Furthermore, we present solutions to Cauchy problems and generalized Cauchy problems for a fractional heat model involving the k -HPFD using the KT transform combined with the Fourier transform. Finally, we propose an integral technique combining the KT and Fourier transforms to construct solutions for the fractional advection-dispersion equation governed by the k -HPFD. The solutions of the Cauchy problems and advection-dispersion equations involving the k -HPFD and its Caputo form are expressed in terms of a generalized Mittag-Leffler function through sequential application of integral transform techniques. It is observed that the solutions for the generalized Cauchy problems and advection-dispersion equations involving the k -HPFD operator reduce to those involving the Hilfer fractional derivative, Riemann-Liouville fractional derivative, and Caputo derivative for specific parameter values.

A higher-order time-fractional evolution problems (EPs) with the Caputo time fractional derivative is considered. A weak singularity typically appears close to the initial time ( t=0 ) in this problem’s solution, which reduces the accuracy of conventional numerical methods with uniform mesh. The technique of nonuniform mesh based on the solution’s acceptable regularity is a very efficient way to regain precision. In chemistry, these equations are often used to simulate intricate diffusion processes with memory effects, particularly whenever pattern formation, domain wall propagation in liquid crystals are involved. In the current study, we solve a time-fractional fourth-order partial differential equation with non-smooth solutions using the quintic trigonometric B-spline (QTBS) technique with temporally graded mesh. The stability and convergence of the proposed numerical scheme are discussed broadly, which illustrates clearly how the regularity of the solution and the mesh grading affect the order of convergence of the proposed scheme, allowing one to select the most effective mesh grading. The plots and tabulated results of some test problems are displayed to validate the accuracy and efficiency of the scheme using graded mesh.

This work exhibits a numerical technique for handling the fractional version of convective radial fins with temperature dependent thermal conductivity. For solving fractional order energy balance equation(FEBE), a computational technique namely Jacobi collocation operational matrix method is considered. Jacobi polynomial is used as the base for operational matrix. The operational matrix transforms the FEBE problem into non linear algebraic equations system. To examine this Fractional order system we obtained the approximate solution for FEBE. The solution examined for Vieta Lucas polynomial also. We carried out a comparative analysis of the homotopy perturbation Sumudu transform technique (HPSTM), the homotopy analysis transform method (HATM) and the Adomian decomposition method (ADM). The comparison of solutions with existing literature demonstrates strong agreement. The numerical simulation for temperature are explained graphically. This study offers valuable insights into the operational performance of fins in various applications, including radiators, boilers, refrigeration devices and oil pipelines, among others. The findings can be utilized to radial fins with both temperature-dependent and constant thermal conductivity in thermal design.

The present work deals with steady MHD silver water (Ag-H2O) nanofluid flow and heat transfer across a vertical porous plate involving heat generation/absorption and suction/injection parameter. To investigate the energy equation, we apply the Cattaneo-Christov heat flux theory. The ruling equation of momentum and energy are converted into a system of nonlinear differential equations with the aid of the convenient similarity transformation. The homotopic procedure is implemented to find out the solution. The impact of significant parameters as magnetic parameter, heat generation/absorption parameter, suction/injection parameter, Prandtl number and nanoparticle theromophysical constant on dimensionless flow velocity and temperature study are depicted through the tables and graphs. A great agreement between the results available in literature and suggested technique is achieved. Calculations of skin friction coefficients and Nusselt number are demonstrated via tables. The outcome reveals that the momentum boundary layer thickness shrinks by enhancing the volume of Ag-nanoparticles.

In this article, natural transform of k-Prabhakar integral, k-Prabhakar derivative, k-Hilfer-Prabhakar fractional derivative (k-HPFD) are calculated. In addition, we also obtain the natural transform of regularized versions of k-Prabhakar integral, k-Prabhakar derivative, k-HPFD. Finally, we solve various k-Hilfer-Prabhakar type Cauchy equations via operations of natural and Fourier transforms. The diffusion equations play a key role in oceanography and all models of hydrodynamics. Our new generalized solutions of k-HPFD type Cauchy problems and diffusion models may be used to explore fluid mechanics, ocean engineering, and wave phenomena and so on. The solutions of Cauchy equations and diffusion models considered with k-HPFD operator and its regularized version are computed in a shape of generalized Mittag-Leffler form by subsequent operations of integral transforms.

The aim of this paper is considering the initial value problem for a system of coupled nonlinear sinh-Gordon equations by the association between two regularization methods: filter and truncation Fourier. Firstly, we give an example to show that the problem does not satisfy the third property which is called ill-posed in the sense of Hadamard. Secondly, under some a priori assumptions, we propose the stable regularization methods to regularize the system, i.e. the corresponding regularized solution converge to the exact solution in \(L^2\)-norm. Finally, to illustrate the proposed efficiency in the theoretical part, we show some numerical tests to check the convergence of the regularized solution and the regularized errors.

In this work, we extend the modified homotopy analysis transform method (MHATM) for studying the three different coupled time-fractional physical problems. The first one is coupled time-fractional Whitham-Broer-Kaup (W-B-K) equations, and others are coupled modified Boussinesq equations and coupled approximate long wave equations as the special cases of W-B-K equations. The fractional W-B-K model is a coupled structure that describes the nonlinear evolution of shallow-water waves. The novelty of the proposed algorithm, the fractional derivative, is taken in the Caputo-Fabrizio (CF) sense, which consists of an exponential form of the non-singular kernel. With the aid of Banach's fixed point theory, the uniqueness and convergence analysis for the coupled W-B-K equations is presented through the theorems. With the help of Picard's stable approach, the stability analysis of the proposed technique is shown. We study the comparison for the solutions of MHATM for CF time-fractional derivative with the solutions derived with the aid of other techniques. The main advantage of the MHATM with the assistance of CF derivative is that, it offers solutions to problems in a rapidly convergent series leading to ideal solutions. The accuracy and efficiency of the present method have been shown through different graphical as well as tabulated analyses. However, the results indicate that MHATM with CF fractional derivative is a good organization and applicable to solve highly nonlinear various fractional physical problems like W-B-K equations.

The primary objective of the work presented in this paper is to introduce and explore a new probability distribution function that incorporates incomplete ℵ -functions. These functions play a crucial role in the formulation of this distribution, allowing for a broader range of applications in statistical modeling. In addition to introducing the new distribution, the paper discusses several essential mathematical concepts that underpin its formulation. Among these, the Mellin and Laplace transforms are highlighted, as they provide significant tools for analyzing the properties of the newly defined distribution. These transforms facilitate the derivation of various statistical measures and characteristics, which are crucial for understanding the behavior of the distribution. Furthermore, the paper delves into the properties of the inverse Gaussian distribution as it relates to the incomplete ℵ -function. This examination not only illustrates the connection between the new distribution and existing statistical models but also showcases the potential applications of these functions in probability theory and related fields. Overall, this work aims to enrich the study of probability distributions by integrating incomplete ℵ -functions, thereby offering new avenues for research and practical applications in statistical analysis.

Applying an approximate numerical technique to a system of coupled nonlinear ordinary differential equations (ODEs) within a supply chain model is particularly valuable due to its relevance to industrial operations. This paper examines and contrasts two algorithms designed to solve a supply chain model incorporating a control variable and time-fractional derivatives. The first method utilizes the Legendre spectral collocation method (LSCM) combined with Caputo fractional derivatives of arbitrary order. This approach transforms the model into a series of algebraic equations. The second method is based on the fundamental principles of fractional calculus and employs the Newton polynomial interpolation (NPI). Both methods are used to generate numerical solutions for the fractional supply chain model. The effectiveness and accuracy of these computational results are evaluated and compared through various figures and tables presented in the paper.

The present genome-wide association studies (GWAS) were conducted on the Badri breed of indicine cattle, a breed renowned for its disease resistance and adaptability to the challenging terrain of the Himalayan mountains. The milk production of Badri cattle is lower as compared to indicine and taurine dairy cattle breeds. However, contemporary techniques, particularly the GWAS, offer opportunity for systematic improvement in milk production and fertility of indicine cattle by identifying genomic variations associated with production and reproduction traits. The DNA samples of 96 Badri cows were genotyped using a double digestion restriction associated DNA sequencing approach. Subsequently, a standardized bioinformatics pipeline was employed to identify variants, wherein only single-nucleotide polymorphisms (SNPs) meeting stringent quality control criteria were retained for further analysis. Following filtration, a total of 65,483 high-confidence SNPs were utilized to conduct GWAS on production traits (including lactation length, peak yield, total milk yield, and days to attain peak yield within lactation) and reproduction traits (encompassing dry period and calving interval). Various genomic regions and candidate genes (LOC785621, ADCK1, DIP2C, WDR26, LOC510860, RPS6KC1, KCTD1, GPC3, SLC16A2, TIPRL, DYNC1I1, GRM1, PBX3, AKR1C4, SRGAP2, RPS6KC1, bPLP-II, NRG1 and TENM1) and their enrichment pathways were identified which may have a potential role in regulating milk production and female fertility in Badri breed of indicine cattle. The identified genomic regions will aid in early and best individual selection of cattle for higher milk production, improved fertility and in the long term, this may assist to shape future genetic improvement efforts of indicine cattle.

This work presents a numerical approach for handling a fractional Lienard equation (FLE) arising in an oscillating circuit. The scheme is based on the Vieta Lucas operational matrix of the fractional Liouville-Caputo derivative and the collocation method. This methodology involves a systematic approach wherein the operational matrix aids in expressing the fractional problem in terms of non-linear algebraic equations. The proposed numerical approach utilizing the operational matrix method offers a vital solution framework for efficiently tackling the fractional Lienard equation, addressing a key challenge in mathematical modeling. To analyze the fractional order system, we derive an approximate solution for the FLE. The solutions are explained graphically and in tabular form.

In this paper, we present a Wolfe-type dual model containing the Caputo-Fabrizio fractional derivative, weak and strong duality results, number of Kuhn-Tucker type sufficient optimality conditions and duality results for variational problems (VPs) with Caputo-Fabrizio (CF) fractional derivative operator under weaker invexity assumptions. This newly developed fractional derivative operator delivers an exponential type kernel of nonsingular nature which characterizes the dynamics of physical systems and engineering processes with memory characteristics in a better way. This derivative operator is a convolution of first-order derivative and an exponential function. The proposed work also derives the global optimality criterion of the primal problem, Mond-Weir type duality results, and Mangasarian type strict converse duality theorem in view of this fractional differential operator possessing an exponential type kernel. The derived theorems investigate the global optimal solution of the primal problem. The main results of the present work are duality theorems and sufficient optimality conditions for VPs possessing the CF derivative. In view of applications of above derived optimality theorems, Mond-Weir type duality results have been established subjected to invexity assumptions. These applications and results generalize other important duality results of VPs and also provide results connected to duality with generalized invexity in mathematical programming. Several conventional results can also be seen as a special case of the obtained results in this work. 2010 Mathematics Subject Classification 90C29; 90C46; 26A33

This paper examined the features of an infection therapy for fractional-order quarry-hunter systems in order to control sickness. It focused especially on how illnesses and several populations combine to affect how well harvesting policies work. We created a new dynamic model full of such ideas by examining systems with fractional-order non-integer systems and introducing fractional-order systems that can remember in order to comprehend that specific system. These thresholds are essential for directing management strategies, according to research on the presence, uniqueness, and stability of solutions to these models. Additionally, we presented particular MATLAB-based numerical methods for fractional-order model. Through a series of numerical application experiments, we validated the method's efficacy and its value in guiding strategy modifications regarding harvesting rates in the face of epidemic infections. This demonstrates the necessity of using a fractional approach in ecosystem research in order to improve the methods used for resource management. This paper primarily focused on the unique insight brought into the quarry-hunter models with infectious diseases by the fractional-order dynamics in ecology. The results are meaningful especially since they can be utilized to come up with effective measures to control diseases and even promote the sustainability of ecological systems.

This article investigates the Poisson equation inverse source problem. This is an ill-posed, i.e. a small change in the data will lead to a very large change in the solution. Therefore, a regularized solution is necessary. In this work, we construct the regularized solution by truncation method. We also investigate the convergent rate between the regularized solution and the sought solution in \(L^j(0,\pi)\).

In this paper, a fractional food chain system consisting of a Holling type Ⅱ functional response was studied in view of a fractional derivative operator. The considered fractional derivative operator provided nonsingular as well as a nonlocal kernel which was significantly better than other derivative operators. Fractional order modeling of a model was also useful to model the behavior of real systems and in the investigation of dynamical systems. This model depicted the relationship among four types of species: prey, susceptible intermediate predators (IP), infected intermediate predators, and apex predators. One of the significant aspects of this model was the inclusion of Michaelis-Menten type or Holling type Ⅱ functional response to represent the predator-prey link. A functional response depicted the rate at which the normal predator consumed the prey. The qualitative property and assumptions of the model were discussed in detail. The present work discussed the dynamics and analytical behavior of the food chain model in the context of fractional modeling. This study also examined the existence and uniqueness related analysis of solutions to the food chain system. In addition, the Ulam-Hyers stability approach was also discussed for the model. Moreover, the present work examined the numerical approach for the solution and simulation for the model with the help of graphical presentations.

A new type of methodology termed the local fractional natural homotopy perturbation method (LFNHPM) with the local fractional derivative operator (LFDO) was implemented in this study. The hybrid methodology combines the natural transform method (NTM) with the homotopy perturbation method (HPM).To validate and illustrate the efficacy of the current method, two challenges are solved. The results obtained using the LFNHPM show excellent agreement with the LFVIM and LFRDTM, demonstrating that the LFNHPM is an effective approach for obtaining the approximate and closed-form solutions of fractional models. We established that our approach for fractional models is accurate and straightforward and researcher can use this approach to solve various problems.

Genome-wide association studies (GWAS) offer potential for discovering genomic regions that can be exploited to increase milk production. However, available GWAS and single nucleotide polymorphism (SNP) datasets are heavily skewed towards taurine breeds, which restricts their utility for genomic research in indicine cattle breeds. This study conducts a GWAS on the Badri breed of Indicine cattle to estimate variance components and identify significant variants associated with milk composition traits, utilizing double digest restriction-site associated DNA (ddRAD) sequencing data. A total of 65,483 high-confidence SNPs were identified and utilized to conduct GWAS on various milk composition traits, including fat percent (FP), protein percent (PP), casein percent (CP), lactose percent (LP), glucose percent (GP), galactose percent (GLP), total solids percent (TS), and solids-not-fat percent (SNF), each analysed separately. The heritability estimates for the studied milk composition traits were 0.386 for fat percent (FP), 0.427 for protein percent (PP), 0.469 for casein percent (CP), 0.567 for lactose percent (LP), 0.547 for glucose percent (GP), 0.590 for galactose percent (GLP), 0.437 for total solids percent (TS), and 0.476 for solids-not-fat percent (SNF). Several genomic regions and candidate genes, including SLC9A9, LPP, C2H2orf76, LGSN, HMGCS2, Bv1, SCYL2, PLAC8, SRGAP2, CR2, ZNF787, OTUB2, DSC2, SYNPO2, and CTNNA3 which may have a potential role in regulating milk production in indicine cattle were identified. The high confidence SNPs and candidate genes will be an important inclusion into commercial genotyping arrays for the early and best selection of breeding animals for desired milk composition and improved production.

The present study was carried out on 96 animals representing three distinct colour variants of Badri cattle to investigate the genetic diversity, population structure and substitution mutations in the genetic codons due to single nucleotide variations. The DNA samples of 96 Badri cows were genotyped using a double digestion restriction associated DNA (ddRAD) sequencing approach. A standardized bioinformatics pipeline was employed to identify single nucleotide polymorphisms (SNPs), initially detecting 7,168,552 SNPs through alignment with the Bos indicus reference genome assembly. Subsequent stringent quality filtration yielded 65,483 high-confidence SNPs for downstream analysis. Genetic diversity analysis of the Badri cattle population resulted in average values of 0.145, 0.088, and 0.091 for Shannon’s diversity Index (I), Simpson’s Diversity (h), and Simpson’s Unbiased Diversity (uh), respectively. Genetic similarities between the black and brown, black and grey, and brown and grey Badri variants were found to be 0.9972, 0.9980 and 0.9970, respectively. Tajima’s D diversity value was observed to be significant and positive for 99.29% of high-confidence SNPs (65,483). STRUCTURE analysis showed admixture among the three Badri colour variants, suggesting a lack of genetic differentiation. Annotation of high-confidence SNPs regarding genetic codon changes indicated maximum substitutions in the GGC with GGT (22 occurrences), followed by AAC to AGC (20 occurrences), GAA to TAA (19 occurrences) and CAA to CAG (19 occurrences). The study concludes there are genetic similarities among colour variants, lack of rare alleles, balancing selection, sudden population contraction and genetic codon substitutions within the Badri cattle population. Insights derived from SNP data analysis hold potential significance for conservation initiatives and breed improvement programs for indicine cattle.