Integral Transform Research Articles

A New General Integral Transform of Modified Laguerre’s Polynomial 𝐿𝐚,𝐛,𝐜,𝐧(𝑡)

Objectives: The core objective of this paper is to introduce the explicit Modified Laguerre polynomial 𝐿𝑎,𝑏,𝑐,𝑛 (𝑡) and obtain a new general integral transform of Modified Laguerre’s Polynomial, by using this general transform, some transforms of Laguerre’s Polynomial are identified as special cases. Method: We derive transforms of the polynomials 𝐿𝑎,𝑏,𝑐,𝑛 (𝑡), 𝐿𝑛 (𝑡) by using this new integral transform approach and found the transforms like Laplace, 𝛼-Laplace, Sawi, Elzaki, Sumudu, Natural, Aboodh, Pourreza, Mohand, G_transform, Kamal transforms and their relationship for the same. Findings: A new general integral transform of the Modified Laguerre’s Polynomial 𝐿𝑎,𝑏,𝑐,𝑛 (𝑡) has been obtained, and a special case for the polynomial 𝐿𝑛 (𝑡) has been derived by using this transform. Novelty: This is a novel general integral transform for this polynomial and by applying this transform to the Modified Laguerre’s polynomial as special transform cases like Laplace, 𝛼-Laplace, Sawi, Elzaki, Sumudu, Natural, Aboodh, Pourreza, Mohand, G_, Kamal, Sumudu, Elzaki, Natural, and Pourreza are obtained. Keywords: Modified Laguerre polynomial; Laguerre polynomial; A New Integral Transform; Laplace Transform; Sumudu Transform

Photothermal impacts induced by laser pulse in a 2D semiconducting medium with temperature-dependent properties under strain–temperature rate-dependent theory

The purpose of this research is to introduce a new model for studying the photothermal process in a 2D semiconducting material with temperature-dependent properties using the recently modified Green and Lindsay theory. The surface absorption fashion generated by a laser pulse is used to heat a medium whose surface is subjected to cooling impact and considered traction free. The integral transformation method is used to attain a general solution via Fourier and Laplace transforms, and the inverse Laplace transform is performed numerically using Riemann sum approximation. A silicon element is utilized to get findings that are graphically displayed as an application in which the consistency of outcomes may be deduced.

Far-Zone Effects for Spherical Integral Transformations II: Formulas for Horizontal Boundary Value Problems and Their Derivatives

AbstractIntegral formulas represent a methodological basis for the determination of gravitational fields generated by planetary bodies. In particular, spherical integral transformations are preferred for their symmetrical properties with the integration domain being the entire surface of the sphere. However, global coverage of boundary values is rarely guaranteed. In practical calculations, we therefore split the spherical surface into a near zone and a far zone, for convenience, by a spherical cap. While the gravitational effect in the near zone can be evaluated by numerical integration over available boundary values, the contribution of the far zone has to be precisely quantified by other means. Far-zone effects for the isotropic integral transformations and those depending on the direct azimuth have adequately been discussed. On the other hand, this subject has only marginally been addressed for the spherical integral formulas that are, except for other variables, also functions of the backward azimuth. In this article, we significantly advance the existing geodetic methodology by deriving the far-zone effects for the two classes of spherical integral transformations: (1) the analytical solutions of the horizontal, horizontal–horizontal, and horizontal–horizontal–horizontal BVPs including their generalisations with arbitrary-order vertical derivative of respective boundary conditions and (2) spatial (vertical, horizontal, or mixed) derivatives of these generalised analytical solutions up to the third order. The integral and spectral forms of the far-zone effects are implemented in MATLAB software package, and their consistency is tested in closed-loop simulations. The presented methodology can be employed in upward/downward continuation of potential field observables or for a quantification of error propagation through spherical integral transformations.

Analytical simulation of the simultaneous adsorption process of methane and water vapor in shales

ABSTRACT Investigations on the diffusion – adsorption of methane and water vapour in shale nanopores are essential for developing shale gas extraction. A kinetic model considering diffusion and different adsorption mechanisms was proposed. An analytical solution to the model was obtained through the integral transformation method and used to predict the distributions of methane and water within a shale particle. The results suggest that the reversible adsorption caused by the dispersion force is not the only adsorption mechanism; irreversible adsorption caused by hydrogen bonding or other types of intermolecular forces is more prominent, lengthening the delay to reach equilibrium. The presence of water strongly affects shale, considerably decreasing the diffusivity and adsorption capacity as well as the swelling of the shale matrix caused by the increase in the methane pressure. The diffusion and adsorption process expands to the shale particle surface after the pores near the particle centre are filled. The transport and storage of water vapour are more sensitive to the pore structure than those of methane. In the initial stage of the methane diffusion‒adsorption process, a large number of water molecules enter the particles, and the adsorbed water hinders methane molecules from contacting the shale surface.

Extended material requirement planning (MRP) within a hybrid energy-enabled smart production system

A smart production system can be made energy-efficient using renewable energy and is considered to maintain the extended material requirement planning under a logistics system by using radio frequency identification. The tracking technology provides information about products with real-time notification. This study investigates renewable energy usage within a smart production system as renewable energy can contribute to Net Zero Emissions. The logistics framework involves an autonomation technology-based production system, optimum cash flow, logistics, and carbon emissions. Time is an essential influencer for material requirement planning. The model is solved with a Laplace integral transformation, where an associated matrix method is utilized by the input–output analysis. The theoretical concept is elaborated through an illustrative numerical example, where the energy consumption and corresponding net present values are evaluated. Numerical and graphical studies prove the effectiveness of the model for the use of renewable energy within for material planning under a reverse logistics system. The result reveals that efficient renewable energy consumption can save considerable costs and reduce the negative net present value of the system. It is found that skilled workers are worthy of a smart production system, not only in a qualitative aspect but also in an economic aspect.

Photometric redshifts probability density estimation from recurrent neural networks in the DECam local volume exploration survey data release 2

Photometric wide-field surveys are imaging the sky in unprecedented detail. These surveys face a significant challenge in efficiently estimating galactic photometric redshifts while accurately quantifying associated uncertainties. In this work, we address this challenge by exploring the estimation of Probability Density Functions (PDFs) for the photometric redshifts of galaxies across a vast area of 17,000 square degrees, encompassing objects with a median 5σ point-source depth of g= 24.3, r=23.9, i= 23.5, and z= 22.8 mag. Our approach uses deep learning, specifically integrating a Recurrent Neural Network architecture with a Mixture Density Network, to leverage magnitudes and colors as input features for constructing photometric redshift PDFs across the whole DECam Local Volume Exploration (DELVE) survey sky footprint. Subsequently, we rigorously evaluate the reliability and robustness of our estimation methodology, gauging its performance against other well-established machine learning methods to ensure the quality of our redshift estimations. Our best results constrain photometric redshifts with the bias of −0.0013, a scatter of 0.0293, and an outlier fraction of 5.1%. These point estimates are accompanied by well-calibrated PDFs evaluated using diagnostic tools such as Probability Integral Transform and Odds distribution. We also address the problem of the accessibility of PDFs in terms of disk space storage and the time demand required to generate their corresponding parameters.We present a novel Autoencoder model that reduces the size of PDF parameter arrays to one-sixth of their original length, significantly decreasing the time required for PDF generation to one-eighth of the time needed when generating PDFs directly from the magnitudes.

Comprehensive two-dimensional analytical modeling of groundwater levels in bi-directional sloping heterogeneous aquifers under variable recharge conditions

This paper develops a comprehensive two-dimensional (2D) groundwater model that surpasses traditional one-dimensional approaches by incorporating extensive hydrological and geological data typically acquired through well drilling. The primary objective of this research is to explore and characterize the complex dynamics of groundwater flow within sloping heterogeneous aquifers subject to rainfall-induced recharge events. To achieve this, the study utilizes the 2D Boussinesq equation, which is enhanced with meticulously defined boundary conditions to more accurately reflect real-world scenarios. The Integral Transform Technique (ITT) is employed to derive an analytical solution that integrates the effects of variable rainfall recharge, aquifer inclination angles, and inherent heterogeneity. This analytical model effectively captures varying recharge dynamics, both spatially and temporally, contributing to a better understanding and management of groundwater systems. The paper presents a new analytical framework, offering deeper insights into how natural factors impact porous medium behavior, thereby providing strategic references for sustainable water resource management.

Solving Undamped and Damped Fractional Oscillators via Integral Rohit Transform

Background: The dynamics of fractional oscillators are generally described by fractional differential equations, which include the fractional derivative of the Caputo or Riemann-Liouville type. These equations induce classical oscillator equations like the harmonic oscillator equation, to include fractional order derivatives. Solving fractional differential equations numerically can be challenging due to the non-local nature of fractional derivatives. Objective: In this paper, a recently developed integral Rohit transform is utilized for solving systems of undamped and damped fractional oscillators characterized by differential equations of fractional or non-integral order involving the Caputo-fractional derivative operator. The solutions of fractional systems which include undamped-simple fractional oscillators, undamped-driven fractional oscillators, damped-driven fractional oscillators, and damped-fractional oscillators are obtained. Methods: by applying the integral Rohit transform, also written as RT. Differential equations of fractional or non-integral order are generally solved by utilizing methods which include the fractional variational iteration approach, the homotopy-perturbation method, the equivalent linearized method, the Adomian decomposition method, etc. Results: This paper demonstrates the effectiveness, reliability, and efficiency of the integral Rohit transform in solving fractional systems, which include undamped-simple fractional oscillators, undamped-driven fractional oscillators, damped-driven fractional oscillators, and damped-fractional oscillators and are characterized by differential equations of fractional or non-integral order involving the Caputo-fractional derivative operator. Conclusions: The Rohit transform brought the progressive principles or methodologies that offer new insights or views on the problems examined in the paper, distinguishing itself from existing methods and doubtlessly beginning up new research instructions. It provided precise results for the specific problems discussed in the paper, surpassing the capabilities of other methods in terms of decision, constancy, or robustness to noise and disturbances.

Three-dimensional solute transport in finite and curved porous media with surface input sources

In this paper, an analytical solution for three-dimensional solute transport in porous media between two curved surfaces is investigated. It is assumed that the groundwater velocity and dispersion coefficient vary with time and position. Groundwater velocity is not considered to be horizontal. The components of dispersion coefficient along the axes are considered to be proportional to the square of corresponding the position variable. The dispersion coefficient components along axes are proportional to the corresponding component of groundwater velocity in temporal aspects while former is squarely proportional to letter one in position components. It is assumed that the sources originate from two curved surfaces. The nature of the source on the two surfaces is the same, but there may be a variation in potential. Initially, the aquifer's domain is supposed to be uniformly polluted. The Laplace Integral Transformation Technique (LITT) is used to obtain analytical solutions. Numerical examples are given to demonstrate the effects of various factors on the solute concentration profile in a system where advection and dispersion play important roles.In addition, the sub-case of horizontal flow is also discussed. The model is extremely useful in analyzing and dealing with widespread surface sources of groundwater pollution in simulated agricultural fields or urban dumping areas.

THERMAL BEHAVIOUR OF A CIRCULAR PLATE UNDER CAPUTO-FABRIZIO FRACTIONAL IMPACT WITH SECTIONAL HEATING

Recent advances in the understanding of the precise physical thermal behaviour of various solids under the effect of fractional-order derivatives have boosted the study of thermoelasticity, which is primarily important in various industrial designs of usable structural materials. We investigated a thin circular plate that was subjected to additional sectional heating on its top and lower surfaces while creating thermal insulation around its outside border. In this work, we maintained the heat transfer equation while accounting for the impact of Caputo-Fabrizio fraction-order derivatives. According to specified boundary constraints, the integral transformation approach is used to assess the analytical solution of the displacement, temperature change, and thermal stresses. Furthermore, various functions and fractional parameters are computed using the material properties of aluminium metal plates for numerical purposes.

Special functions with general kernel: Properties and applications to fractional partial differential equations

Abstract In this paper, we reconstruct the gamma and beta functions using a general kernel function in their integral representations. We also reconstruct the Gauss and confluent hypergeometric functions using the beta function with general kernel in their series representations. The general kernel function we use here can be chosen as any special function such as exponential function, Mittag-Leffler function, Wright function, Fox-Wright function, Kummer function or M-series. Using different choices of this general kernel function, various of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions in the literature can be obtained. In this paper, we first obtain the integral representations, functional relations, summation, derivative and transformation formulas and double Laplace transforms of the special functions we construct. Furthermore, we compute the solutions of some fractional partial differential equations involving special functions with general kernel via the double Laplace transform and graph some of the solutions for specific values. Finally, we obtain the incomplete beta function with general kernel by defining the beta distribution with general kernel.

A universal application of the general integral transform: New technique for the analysis of nonlinear oscillator systems

The wide applicability of a nonlinear oscillator is the main reason why it has gained so much attention from mathematicians and scientists, while it is difficult to be solved both numerically and theoretically. Variational iteration method is one of the numerical approaches which are used to investigate these nonlinear systems. In this paper, we present a nonlinear oscillator arising in a micro-electromechanical system as an example and derive its analytical solution by He’s frequency formula method and variational iteration method coupled with a new general integral transformation. The Lagrange multiplier is formulated and the new iterative format for the correction functional of the variational iteration method is given by the general integral transformation. The approximate nonlinear frequencies are achieved by this new technique and compared with the numerical ones obtained by the Runge–Kutta method. The new general integral transformation has the same essential qualities as those for Laplace and Fourier transform, and it becomes a promising tool to nonlinear problems when coupled with the variational iteration method.

Kernel Function, $q$-Integral Transformation and $q$-Heun Equations

We find kernel functions of the $q$-Heun equation and its variants. We apply them to obtain $q$-integral transformations of solutions to the $q$-Heun equation and its variants. We discuss special solutions of the $q$-Heun equation from the perspective of the $q$-integral transformation.

Channel flow dynamics of fractional viscoelastic nanofluids in molybdenum disulphide grease: A case study

Nanoscopic fluids especially viscoelastic Nanofluids are very useful in engineering and industrial problems. This research is to determine the open channel flow of a viscoelastic nanofluid (namely Oldroyd-B (OBNF)). Oldroyd-B fluid (OBF) was used as the base fluid and molybdenum disulphide nanopatrials were added in the fluid to form desired OBNF. To convert the mathematical model to a fractional model from a classical order partial differential equation PDE, fractional type derivative named Caputo-Fabrizio (CF) was used. The main objective of this study is to find the exact mathematical solution for the temperature, concentration and velocity distributions by using integral transformation technique. Final results are discussed graphically for the influence of different parameters on calculated temperature, concentration and velocity. Skin friction of the said fluid and engineering related dimensionless numbers including Reynolds number (Re), Prandtl number (Pr), Grashof number (Gr) and Schmidt number (Sc) are also discussed. At the end a comparison is illustrated in graphical form between current studied fluid (i.e. OBNF), another non-Newtonian fluid (i.e. Maxwell Nanofluid (MWNF)) and Newtonian fluid. As we know speed of Newtonian fluid is greater than non-Newtonian fluid, the same statement is validated by our solution. It is also noted that adding molybdenum disulphide nanoparticles to grease, heat transmission increased to 19.11% and mass transmission decreased to 2.51%.

On the analytical solution of the one-dimensional convection–conduction equation for packed-bed thermal energy storage systems

Temperature distribution modeling within packed-bed thermal energy storage (PBTES) systems is crucial to simulate its integration into heat sources and perform techno-economic analyses to assess the actual benefits associated with its use. This article proposes a one-dimensional convection–conduction equation to model a fluid–solid system by assuming volume-averaged properties for the energy balance and determines the analytic solution through Integral Transforms. The present study analyzes the applicability of this analytic solution considering different operational conditions of PBTES systems. The article revealed that the Péclet number (Pe) and the fluid-to-solid capacity ratio (κ) must be limited to obtain stable solutions, while the dimensionless time τ cannot be arbitrary despite computing an analytic solution. A sensitivity study of the solution for parameter a=κPe/2 defined the minimum dimensionless time required for the solution to be stable. This stability was assessed with existing experimental setups, indicating the solution’s feasibility for air–solid PBTES systems.

New Two Parameter Integral Transform "MAHA Transform" and its Applications in Real Life

In this paper, an integral transform with two parameters was proposed, and this transform was applied to obtain the exact solutions for the linear ordinary differential equations with constant coefficients of higher orders. This integral transform was also applied and we called it Maha transform in nuclear physics and for medical applications.

Thermal conductivity impact on MHD convective heat transfer over moving wedge with surface heat flux and high magnetic Prandtl number

The present evaluation focused on heat transfer and magnetic flux along the moving non-conducting wedge shape with thermal conductivity and surface heat flux effects. A suitable and well-known similarity transformation for integration is used for the coupled non-linear partial differential equations with stream formulations. The Keller Box technique is utilized to integrate the final non-similar equations numerically. The discretized algebraic equations are displayed graphically and numerically using the MATLAB software. The physical characteristics like temperature, magnetic field, velocity profiles, skin friction, heat transfer and magnetic intensity are investigated with various factors. The influence of relevant characteristics like thermal conductivity ξ, Prandtl number Pr, wedge parameter β, variable viscosity ε, and Biot number Bi is interpreted graphically and numerically. It is noted that velocity graph effects are declined at lower value of Biot number and enhanced value is obtained at the higher value of Biot number. The fluid temperature with highest thermal slip effects is displayed with minimum value of thermal conductivity but minimum value is obtained at large value of thermal conductivity. The maximum heat flux value is obtained at large value of moving parameter.

Application of the New General Integral Transform for Newton’s Law of Cooling

Application of the New General Integral Transform for Newton’s Law of Cooling

Handling Growth and Decay Problems by the New General Integral Transform

Handling Growth and Decay Problems by the New General Integral Transform

Enhanced synthetic generation of channel state information for millimeter‐wave networks in 5G communication systems

AbstractIn 5G communication systems, millimeter‐wave networks are pivotal, relying heavily on Channel State Information (CSI) for effective user‐to‐base station (BS) transmission. However, the acquisition of genuine CSI data remains a hurdle, often due to the expenses associated with simulations or physical experiments. This paper introduces an innovative method for generating artificial CSI data from real datasets, aiming to closely replicate authentic CSI samples. The procedure begins with an initial clustering analysis, followed using Principal Component Analysis and Uniform Manifold Approximation and Projection to reduce dimensionality. Then, the data distributions are transformed into multivariate normal distributions using Probability Integral Transformations (PIT). For data synthesis, Multilayer Perceptron based regression models are utilized, followed by inverse PIT transformations to return the data to its original space. Our method is compared against KDE‐based algorithms, demonstrating superior fidelity in reproducing real CSI samples. Additionally, we stress the importance of capturing CSI correlations among different BSs to refine data synthesis. This research propels forward data synthesis techniques, offering potential solutions for mitigating interference challenges in dense MMW networks and advancing 5G communication systems.