Maple Code Research Articles

Proficiency of probability distributions in unit hydrograph derivation

Abstract The probability distribution function (PDF)-based unit hydrographs (UHs) are gaining momentum in an application for more accurate rainfall-runoff transformation. Employing seven statistical performance indices, R2, NSE, MSE, RMSE, MAE, MAPE, and SE in generalized reduced gradient nonlinear programming (GRG-NLP) optimization, 18 known and 12 adaptable PDF-based UHs were assessed against UHs derived from 18 storms in 7 basins across the United States, Turkey, and India. To this end, 27 Maple codes were proposed for UH-application requiring only peak discharge (qp), time to peak (tp), and time base (tb) for derivation. The introduced PDFs, such as Dagum, Generalized Gamma, Log-Logistic, Gumbel Type-I, and Shifted Gompertz, replicated the observed data-derived UHs more closely than did the known PDFs, like Inverse Gaussian, two-parameter gamma distribution (2-PGD), Log-Normal, Inverse-Gamma, and Nagakami. Among the three-parameter (6 nos.), two-parameter (21 nos.), and single-parameter (3 nos.) PDFs, the Dagum, Log-Logistic, and Poisson consistently outperformed their respective counterparts in replication.

Stagnant points and uniform film analysis of Eyring fluid film flow on a vertically upward moving slippery flat plate

In this paper, we analyze how Eyring fluid film behavior on a vertically upward moving slippery flat plate can help predictive models in engineering, notably in coating and lubrication operations. This paper deals with the stagnant points and uniform film analysis of Eyring fluid film flow on a vertically upward moving slippery flat plate. The formulated ordinary differential equation is solved for exact analytic solution. Exact analytic expressions for velocity, flow rate, average velocity, shear stress components, and stagnant points are derived. A highly nonlinear algebraic equation is derived for film thickness. Equation is solved by Newton’s method using Maple code. The thickness of film widens with increasing relaxation time, constant plate velocity, and fluid density, while it decreases with increasing fluid viscosity and constant slip parameter. The analysis delineates that the positions of stagnant points tend to relocate closer the slippery plate as the Stokes number, Deborah number, and constant slip parameter increase. A high Deborah number enhances drainage, causing stagnant points to relocate closer to the slippery plate and the development of a stable elastic layer adjacent to the plate. For stagnant points and fluid film thickness, a comparison between the Eyring fluid and existing studies (EPTT, LPTT, UCM, and Newtonian fluid) is also provided. The validity of this paper with the existing studies is also presented by reducing Eyring fluid model to Newtonian model. The results of this research are significant for a wide range of biofluid applications, including agrochemical uses, paint and surface coating flow behavior, thin films on the cornea and lungs, and chemical and nuclear reactor design.

The function of relaxation and retardation phenomena on time-dependent natural convective and oscillatory flowing of dissipative Oldroyd-B fluid via an infinite vertical plate with fixed suction

Constant suction phenomenon has been executed in free convective Oldroyd-B material in an oscillatory flow mode over infinite vertical porous plate. For heat transfer, this study includes the influence of heat dissipation. The resulting governing equations are transferred into a set of dimensionless form of coupled-ordinary differential equations with appropriate non-dimensional variables. MAPLE code is developed for the regular perturbation method and executed to solve the set of non-dimensional couple-ordinary derivative equations. The impact of different embedded parameters on the mean, fluctuating, transient velocities and temperatures on the Oldroyd-B fluid have been discussed and depicted in graphical forms. The influence of some physical terms on the plate mean friction and plate heat mean gradient are computed and demonstrated in a tabular form. The attained results show that as an increase in relaxation time, greater amplitude of temperature fluctuations is observed. Remarkable enhancement is observed in all the parameters (Grashof, Prandtl and Eckert numbers) for transient temperature profiles.

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

The Mechanics Difference Between the Outer Torus and Inner Torus

AbstractThe formulation used by the most of studies on elastic torus are either Reissner’s mixed formulation or Novozhilov’s complex-form one; however, for vibration and some displacement boundary-related problem of torus, those formulations face a great challenge. It is highly demanded to have a displacement-type formulation for torus. In this article, we will carry on the first author’s previous work (Sun, 2010, “Closed-Form Solution of Axisymmetric Slender Elastic Toroidal Shells,” J. Eng. Mech., 136, pp. 1281–1288.), and with the help of our own maple codes, we are able to simulate some typical problems of torus. The numerical results are verified by both finite element analysis and H. Reissner’s formulation. Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio. The analysis of a torus must be done by using the bending theory of a shell instead of membrane theory of shells, and also reveal that the inner torus is stronger than outer torus due to their Gaussian curvature. One of the most interesting discovery is that the crowns of a torus, the turning point of the Gaussian curvature at ϕ = 0, π, are the line where the mechanics response of inner and outer torus is almost separated.

Improper integral with exponential function

In this paper, we explore the improper integral with exponential function f = xx is approached to infinite series, and also prove the convergence of these series. An improper integral converges if the limit defining it exists. We use Maple code to calculate the infinite series. The application of improper integral appear in several domain in science. As an application in this paper, three examples are given to illustrate the effectiveness of our main result.

Some Single Valued Neutrosophic Queueing Systems with Maple Code

Some Single Valued Neutrosophic Queueing Systems with Maple Code

Some Convolution Identities for Catalan Numbers

In the paper, we propose a series of convolution identities for the sequence of Catalan numbers. The proofs of two of them are provided and the other identities are proposed as conjectures. We include a simple Maple code to generate and confirm the identities.

Aerodynamic shape and drag scaling law of a flexible fibre in a flowing medium

The study of a flexible body immersed in a flowing medium is one of the best way to find its aerodynamic shape. This Letter revisited the problem that was first studied by Alben et al. (Nature 420, 479–481, 2002). To determine the aerodynamic shape of the fibre, a simpler approach is proposed. A universal drag scaling law is obtained and the universality of the Alben-Shelley-Zhang scaling law is confirmed by using dimensional analysis. A complete Maple code is provided for finding aerodynamic shape of the fibre in the flowing medium.

Using Stochastic SIS Epidemic Model with Markovian Environment for Infectious Diseases

There are various types of stochastic epidemic models that can be formulated to deal with different types of diseases. In this project (SIS) epidemic model Susceptible - Infected- Susceptible will be considered relating to the Continuous Time Markov chains process. Each type of the epidemic models studies the disease according to its status. In particular, the SIS epidemic model regards infectious diseases. The Maple code is provided in this project, which is created to produce and predict the number of infected individuals and the disease prevalence at a fixed time.

Elliptic Stable Envelopes of Affine Type A Quiver Varieties

Abstract We generalize Smirnov’s formula for the elliptic stable envelopes of the Hilbert scheme of points in $\mathbb {C}^2$ to the case of affine type $A$ Nakajima quiver varieties constructed with positive stability condition. We allow for arbitrary choices of polarization and a fairly general choice of chamber. This paper is a companion to the Maple code developed by the author, which implements the formulas described in this paper.

Nonlinear elastic deformation of Mindlin torus

The nonlinear deformation and stress analysis of a circular torus is a difficult undertaking due to its complicated topology and the variation of the Gauss curvature. A nonlinear deformation (only one term in strain is omitted) of Mindlin torus was formulated in terms of the generalized displacement, and a general Maple code was written for numerical simulations. Numerical investigations show that the results obtained by nonlinear Mindlin, linear Mindlin, nonlinear Kirchhoff–Love, and linear Kirchhoff–Love models are close to each other. The study further reveals that the linear Kirchhoff–Love modeling of the circular torus gives good accuracy and provides assurance that the nonlinear deformation and stress analysis (not dynamics) of a Mindlin torus can be replaced by a simpler formulation, such as a linear Kirchhoff–Love theory of the torus, which has not been reported in the literature.

A Maple Code to Perform Operations on Single Valued Neutrosophic Matrices

A Maple Code to Perform Operations on Single Valued Neutrosophic Matrices

Similarity solutions of Prandtl mixing length modelled two dimensional turbulent boundary layer equations

The exact similarity solutions of two dimensional laminar boundary layer were obtained by Blasius in 1908, however, for two dimensional turbulent boundary layers, no Blasius type similarity solutions (special exact solutions) have ever been found. In the light of Blasius’ pioneer works, we extend Blasius similarity transformation to the two dimensional turbulent boundary layers, and for a special case of flow modelled by Prandtl mixing-length, we successfully transform the two dimensional turbulent boundary layers partial differential equations into a single ordinary differential equation. The ordinary differential equation is numerically solved and some useful quantities are produced. For numerical calculations, a complete Maple code is provided.

Numerical Solutions of a Heat Transfer for Fractional Maxwell Fluid Flow with Water Based Clay Nanoparticles; A Finite Difference Approach

Fractional-order mathematical modelling of physical phenomena is a hot topic among various researchers due to its many advantages over positive integer mathematical modelling. In this context, the appropriate solutions of such fractional-order physical modelling become a challenging task among scientists. This paper presents a study of unsteady free convection fluid flow and heat transfer of Maxwell fluids with the presence of Clay nanoparticle modelling using fractional calculus. The obtained model was transformed into a set of linear nondimensional, partial differential equations (PDEs). The finite difference scheme is proposed to discretize the obtained set of nondimensional PDEs. The Maple code was developed and executed against the physical parameters and fractional-order parameter to explain the behavior of the velocity and temperature profiles. Some limiting solutions were obtained and compared with the latest existing ones in literature. The comparative study witnesses that the proposed scheme is a very efficient tool to handle such a physical model and can be extended to other diversified problems of a complex nature.

Numerical Exploration via Least Squares Estimation on Three Dimensional MHD Yield Exhibiting Nanofluid Model with Porous Stretching Boundaries

The numerical study of a three-dimensional magneto-hydrodynamic (MHD) Casson nano-fluid with porous and stretchy boundaries is the focus of this paper. Radiation impacts are also supposed. A feasible similarity variable may convert a verbalized set of nonlinear “partial” differential equations (PDEs) into a system of nonlinear “ordinary” differential equations (ODEs). To investigate the solutions of the resulting dimensionless model, the least-square method is suggested and extended. Maple code is created for the expanded technique of determining model behaviour. Several simulations were run, and graphs were used to provide a thorough explanation of the important parameters on velocities, skin friction, local Nusselt number, and temperature. The comparison study attests that the suggested method is well-matched, trustworthy, and accurate for investigating the governing model’s answers. This method may be expanded to solve additional physical issues with complicated geometry.

Hertz elastic dynamics of two colliding elastic spheres

This paper revisits a classic problem in physics — Hertz elastic dynamics of two colliding elastic spheres. This study obtains impact period in terms of hypergeometric function and successfully combines Deresiewicz’s three segmental solutions into one single solution. Our numerical investigation confirms that Deresiewicz’s inversion is a good approximation. As an essential part of this study, a general Maple code is provided.

Small symmetrical deformation of thin torus with circular cross-section

By introducing a variable transformation ξ=12(sinθ+1), a complex-form ordinary differential equation (ODE) for the small symmetrical deformation of an elastic torus is successfully transformed into the well-known Heun’s ODE, whose exact solution is obtained in terms of Heun’s functions. To overcome the computational difficulties of the complex-form ODE in dealing with boundary conditions, a real-form ODE system is proposed. A general code of numerical solution of the real-form ODE is written by using Maple. Some numerical studies are carried out and verified by both finite element analysis and H. Reissner’s formulation. Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell. A general Maple code is provided as essential part of this paper.

Monotonic rising and oscillating of capillary-driven flow in circular cylindrical tubes

Capillary rise is one of the best-known capillarity phenomena, and some capillary flows rise monotonically, whereas others oscillate, but no criteria have been formulated for this scenario. In this paper, the critical radius of the capillary tube is predicted numerically by Levine’s model. To the best of our knowledge, the phase–space diagram of capillary velocity vs height is obtained for the first time, which shows that the phase transition from oscillating to monotonically rising occurs when the phase trajectory decreases exponentially to somewhere other than the “attractor.” Two general Maple codes are provided as an essential part of this paper.

Light from Schwarzschild black holes in de Sitter expanding universe

A new method is applied for deriving simultaneously the redshift and shadow of a Schwarzschild black hole moving freely in the de Sitter expanding universe as recorded by a remote co-moving observer. This method is mainly algebraic, focusing on the transformation of the conserved quantities under the de Sitter isometry relating the black hole co-moving frame to observer’s one. Hereby one extracts the general expressions of the redshifts and shadows of the black holes having peculiar velocities but their expressions are too extended to be written down here. Therefore, only some particular cases and intuitive expansions are presented while the complete results are given in an algebraic code (Cotăescu in Maple code BH01, https://physics.uvt.ro/~cota/CCFT/codes, 2020).