Integral Transform Research Articles

Reduction to master integrals via intersection numbers and polynomial expansions

Intersection numbers are rational scalar products among functions that admit suitable integral representations, such as Feynman integrals. Using these scalar products, the decomposition of Feynman integrals into a basis of linearly independent master integrals is reduced to a projection. We present a new method for computing intersection numbers that only uses rational operations and does not require any integral transformation or change of basis. We achieve this by systematically employing the polynomial series expansion, namely the expansion of functions in powers of a polynomial. We also introduce a new prescription for choosing dual integrals, de facto removing the explicit dependence on additional analytic regulators in the computation of intersection numbers. We describe a proof-of-concept implementation of the algorithm over finite fields and its application to the decomposition of Feynman integrals at one and two loops.

Analytic solutions to the two-dimensional solute advection-dispersion equation coupled with heat diffusion equation in a vertical aquifer section

Analytical and semi-analytical solutions to solute transport equations are not only used to predict the fate and transport of contaminants in soil or groundwater systems, but also to provide reference standards for validation of numerical models. However, when the solute transport equation is coupled with heat, the corresponding analytical solutions are rarely reported, perhaps due to the complexity of the coupled models. The main purpose of this paper is to present an analytical solution to the solute transport equation coupled with heat for the case of modeling solute transport in a vertical section of a homogenous infinite aquifer with steady uniform groundwater flow under constant-flux solute source. In earlier work (Shan and Javandel, 1997), analytical solutions to the solute transport equation in both finite and infinite aquifer thickness were presented, influence of heat (Soret effect) was generally not considered. Heat can, however, have a significant impact on movement of concentration front. In the present work, we first couple the solute equation with a generalized classic heat diffusion equation. To do this, we assume that the heat is not influenced by the solute (Dufour effect) such that only a one-way coupled model is considered. The repeated integral transformation method (RITM) is utilized to obtain the analytical solutions to the proposed coupled model. As a limiting case, a comparison with analytical solution (Shan and Javandel, 1997), equation (21) and other literature findings is done verify the new proposed solutions and validate our results. The influences of the thermopheresis effect and other model parameters are pictured graphically by the use of MATLAB R2015a. In comparison with the earlier work, the present results show that movement of contaminant in the medium is affected by the presence of heat, in particular along the direction of flow, consistent with the literature. The analysis of longitudinal and transverse concentration profiles suggest that the proposed solutions are applicable for monitoring of groundwater contaminants and risk assessment. Also, the solutions should be useful for comparison with numerical models.

Bilateral boundary control of an input delayed 2-D reaction–diffusion equation

In this paper, a delay compensation design method based on PDE backstepping is developed for a two-dimensional reaction–diffusion partial differential equation (PDE) with bilateral input delays. The PDE is defined in a rectangular domain, and the bilateral control is imposed on a pair of opposite sides of the rectangle. To represent the delayed bilateral inputs, we introduce two 2-D transport PDEs that form a cascade system with the original PDE. A novel set of backstepping transformations is proposed for delay compensator design, including one Volterra integral transformation and two affine Volterra integral transformations. Unlike the kernel equation for 1-D PDE systems with delayed boundary input, the resulting kernel equations for the 2-D system have singular initial conditions governed by the Dirac Delta function. Consequently, the kernel solutions are written as a double trigonometric series with singularities. To address the challenge of stability analysis posed by the singularities, we prove a set of inequalities by using the Cauchy–Schwarz inequality, the 2-D Fourier series, and the Parseval’s theorem. A numerical simulation illustrates the effectiveness of the proposed delay-compensation method.

Consolidation behaviors of gas-bearing sediments with modulus varying along depth under horizontal loads

Gas-bearing sediments are widely distributed in marine and lacustrine areas. The engineering properties of gas-bearing sediments, containing enclosed bubbles, are significantly different with that of saturated and unsaturated sediments. This paper investigates the consolidation behaviors of gas-bearing sediments with modulus varying along depth subjected to horizontal load using an extended precise integration method (XPIM). The compressibility of enclosed bubble is introduced to a new derived seepage equation and a varying modulus along depth is considered in governing equations. With the aid of integral transformation and Taylor expansion, such problems are solved by XPIM, which is proved to be markedly efficient and precise for boundary value problems of porous media than traditional numerical methods. Detailed comparisons against analytical solutions are performed to confirm the accurateness of XPIM. Extensive parametric investigations are conducted to examine the influence of saturation degree, varying modulus along depth and type of external load on the consolidation behaviors of gas-bearing sediments. The present work is conveniently utilized to evaluate the behavior of gas-bearing sediments, and subsequently the response of foundations built in gas-bearing soil areas.

Uses of 3D photothermal deflection technique for the investigation of nickel-graphene foam sheets

The aim of this work is to investigate the influence of pores density on the thermal properties of Nickel-graphene foam sheets. The study demonstrates a detailed theoretical and experimental treatment of the three-dimensional photothermal deflection (PTD) technique, under laser excitation of nickel foam of various porosity density coated graphene thin film of 200 nm thickness. Green's function and integral transformations are used in the theoretical treatment. The effect of certain parameters on the sensitivity of the PTD technique with respect to the variation in the specimens' pore density was studied via using analytical and numerical developments. The thermal properties showed that the thermal conductivity and thermal diffusivity were 4.12 W m−1. K −1 and 0.147 cm2. s−1. The numerical results show that decreasing the modulation frequency of the laser causes a decrease in the sensitivity of the thermal signal. Photothermal deflection techniques have shown a good efficiency in determining the thermal properties of porous materials.

Solution to the Unsteady Seepage Model of Phreatic Water with Linear Variation in the Channel Water Level and Its Application

For a semi-infinite aquifer controlled by a river channel boundary, when the Laplace transform is used to solve a one-dimensional unsteady seepage model of phreatic water while considering the influence of the vertical water exchange intensity ε with the change in the river channel water level f(t), a complicated and tedious integral transformation process is required. By replacing f(t) with an operator, this study first derived the analytic formula of the ε term based on the properties of the Laplace transform without the direct participation of f(t) in the transformation. By using f(t) in the form of several types of linear functions, the Laplace transform and inverse transform laws were summarized. The analytical solution to the problem was easily obtained by applying the “integral property” of the transformation to the linear function term with time t. The relative error between the numerical solution and the analytical solution of the example was less than 0.2%, which verified the rationality of the model linearization method and the reliability of the analytical solution. For different boundary conditions, the process of establishing and applying the inflection point method and the curve-fitting method for calculating the model parameters by using dynamic monitoring data for phreatic water is presented with examples.

Editorial for the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”

This editorial text is a short introductory guide to the book edition of the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”, which was published in the MDPI journal Mathematics in the years 2022–2023 [...]

Phase Spaces, Parity Operators, and the Born–Jordan Distribution

Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time–frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (Grossmann in Commun Math Phys 48(3):191–194, 1976. https://doi.org/10.1007/BF01617867), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (Cohen in J Math Phys 7(5):781–786, 1966. https://doi.org/10.1063/1.1931206) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born–Jordan distribution which originates from the Born–Jordan quantization (Born and Jordan in Z Phys 34(1):858–888, 1925. https://doi.org/10.1007/BF01328531). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born–Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born–Jordan distribution.

MATHEMATICAL MODELLING OF DYNAMIC PROCESSES IN A LAYERED INCOMPRESSIBLE HALF-SPACE WITH INITIAL STRESSES UNDER THE ACTION OF A MOVING LOAD

This work is devoted to the study of the influence of the protective coating, initial stresses, mechanical characteristics of materials, movement parameters of the surface load on the stress-deformed state of the elastic base. The relevance of the research results is related to the possibility of their use in the creation of qualitatively new materials, structures and building structures. Two models of a layered incompressible half-space are considered and compared: 1) an elastic plate on an elastic half-space; 2) the top layer (protective coating) is modelled by concentrated masses. The concentrated force moves along the free surface of the protective layer at a constant speed at a certain angle to the surface of the half-space. The solution of the problem was obtained using the method of integral Fourier transformations. Analytical results are given in a general form for materials with an arbitrary elastic potential, for cases of unequal and equal roots of characteristic equations, for various conditions of combination of elements of a layered medium and for any speed of movement of the load. The material with the Bartenev-Khazanovich potential was considered for numerical analysis. The calculations were carried out within the framework of the theory of finite initial deformations. The impact of the moving load, initial stresses and mechanical parameters of the elements of the layered base on the main characteristics of its stress-strain state was studied.

INVERSE PROBLEM FOR A TIMOSHENKO BEAM WITH AN ADDITIONAL VISCOELASTIC SUPPORT UNDER NONSTATIONARY DEFORMATION

The non-stationary loading of a mechanical system consisting of a beam hinged at the edges and an additional support installed in the span of the beam is considered. The deformation of the beam is modeled on the basis of Timoshenko's hypotheses, taking into account the influence of rotatory inertia and shear. The deformation of the beam is described by a system of partial differential equations, which is solved analytically by means of expansion of the unknown functions into the relevant Fourier series and further use of the Laplace integral transformation. It is assumed that the additional support has linear-elastic and linear-viscous components, and the displacements coincide at the point where the additional support is connected to the beam. The reaction between the beam and the additional support is replaced by an external unknown concentrated force applied to the beam, which varies in time. The law of time variation of this unknown reaction is determined by solving the Volterra integral equation. The inverse problem of deformable solid mechanics is solved, that is, it is assumed that the deflection at a point of the beam with the additional support is known, whereas the law of time variation of the external impulse load causing the deflection is unknown. The application point of the external load and the point of the additional support connection are considered to be known and do not change in the process of deformation (when obtaining the solution of the problem it was supposed that these could be any points of the beam except for its ends). The described inverse problem is reduced to a system of two Volterra integral equations of the first kind with regard to the unknowns of the external disturbing load and reaction between the plate and the additional support, which is solved by analytical and numerical method. Analytical relations and calculation results for specific numerical parameters are given. The results obtained in this work can be used for indirect measurement of impulse and shock loads acting on beams with additional supports, for which not only elastic but also linear-viscous characteristics are taken into account.

Generalized Method of Finite Integral Transformations in Linear and Nonlinear Static Problems for Shallow Shells

Generalized Method of Finite Integral Transformations in Linear and Nonlinear Static Problems for Shallow Shells

Interaction among interfacial offset cracks in composite materials under the anti‐plane shear loading

AbstractThe present article considers an anti‐plane stress problem of three cracks at different orthotropic materials' interfaces. According to the geometry of the problem, the governing equations and mixed boundary conditions have been formulated. Fourier integral transformation is used to convert the mixed boundary value problem into dual integral equations, which gives two equations containing infinite series. The investigation of the problem concerning anti‐plane cracks subjected to static loadings is done with the help of the Schmidt method to satisfy the given boundary conditions. The difference in displacements is expanded to proceed further in the problem, which becomes zero outside the cracks. Numerical computations are carried out for the graphical representation of stress intensity factors (SIFs) at all cracks' tips. The Interaction among the cracks as those are in close proximity to each other or move away are represented pictorially. Detailed numerical results and discussion are done for the considered materials, which include aluminium, epoxy and graphite epoxy. The novelty of the present article is the numerical analysis and pictorial presentation of SIFs at the tips of interfacial offset parallel cracks for various crack lengths and normalised heights for different combinations of materials. The authors have obtained variations in SIFs for the cracks at the interfaces of dissimilar composite materials.

Liouville's formulae and Hadamard variation with respect to general domain perturbations

We study Hadamard variations with respect to general domain perturbations, particularly for the Neumann boundary condition. They are derived from new Liouville's formulae concerning the transformation of volume and area integrals. Then, relations to several geometric quantities are discussed; differential forms and the second fundamental form on the boundary.

Some Integral Transform Results for Hilfer–Prabhakar Fractional Derivative and Analysis of Free-Electron Laser Equation

Some Integral Transform Results for Hilfer–Prabhakar Fractional Derivative and Analysis of Free-Electron Laser Equation

Study of Telegraph Equation via He-Fractional Laplace Homotopy Perturbation Technique

A new technique to study the telegraph equation, mostly familiar as damped wave equation is introduced in this study. This phenomenon is mostly rising in electromagnetic influences and production of electric signals. The proposed technique called as He-Fractional Laplace technique with help of Homotopy perturbation is utilized to found the exact and nearly approximated results of differential model and numerical example of telegraph equation or damped wave equation in this article. The most unique term of this technique is that, there is no worry to find the next iteration by integration in recurrence relation. As fractional Laplace integral transformation has some limitations in non-linear terms, to get the result of nonlinear term in this differential mode, He polynomials via homotopy techniques of iteration is proposed to find the result of the computation assignment. The obtained result by this proposed technique directed that this technique is quite ease to apply and convergent rapidly to exact solutions. Numerous examples are described to determine the stability and accuracy of the proposed technique with the graphical explanation.

Classical Solutions of Hyperbolic Equation with Translation Operators in Free Terms

In this paper, we study the question of constructing explicit solutions in a half-space of a hyperbolic equation containing translation operators in space variables in all coordinate directions. Such equations are a natural generalization of classical equations of hyperbolic type, and the resulting solution relates the value of the desired function at different points of the half-space where the process takes place. To construct solutions, a classical operating scheme is used, namely, the formal application of an integral transformation. A theorem is proved that the constructed solutions are classical if the real part of the symbol of the differential-difference operator in the equation is positive. Classes of equations for which this condition is satisfied are given.

Coupled Fixed Point and Hybrid Generalized Integral Transform Approach to Analyze Fractal Fractional Nonlinear Coupled Burgers Equation

In this manuscript, the nonlinear Burgers equations are studied via a fractal fractional (FF) Caputo operator. The results of coupled fixed point theorems in cone metric space are used to discuss the uniqueness of solution to the proposed coupled equations. The solution of the proposed equation is computed via Natural transform associated with the Adomian decomposition method (NADM). The acquired results are graphically presented for some values of fractional order and fractal dimensions. The accuracy and consistency of the applied method is verified through error analysis.

Non-Fourier Heat Conduction of Nano-Films under Ultra-Fast Laser.

The ultra-fast laser heating process of nano-films is characterized by an ultra-short duration and ultra-small space size, in which the classical Fourier law based on the hypothesis of local equilibrium is no longer applicable. Based on the Cattaneo-Vernotte (CV) model and the dual-phase-lag (DPL) model, the two-dimensional analytical solutions of heat conduction in nano-films under ultra-fast laser are obtained using the integral transformation method. The results show that there is a thermal wave phenomenon inside the film, which becomes increasingly evident as the elapse of the lag time of the temperature gradient. Moreover, the wave amplitude in the vertical direction is much larger than that in the horizontal direction of the nano-film. By comparing the numerical result of the two models, it is found that the temperature distribution inside the nano-film based on the DPL model is gentler than that of the CV model. Additionally, the temperature distribution in the two-dimensional solution is lower than that in the one-dimensional solution under the same Knudsen number. In the comparison results of the CV model, the maximum peak difference in the thermal wave reaches 75.08 K when the Knudsen number is 1.0. This demonstrates that the horizontal energy carried by the laser source significantly impacts the temperature distribution within the film.

A study of double general transform for solving fractional partial differential equations

In this research, we introduce the basic properties of the single and double general transforms. New interesting results related to fractional operators in the sense of Caputo derivative are investigated and proved. In this study, the general double transformation is utilized to study the application on some special functions, such as the Mittag–Leffler function, and on the Caputo's fractional derivative of different and higher orders. Moreover, we establish new interesting formulas and prove some theories that can save efforts and time for researchers as they investigate the properties and applications of integral transformations on fractional operators. The results obtained in this research are tested by solving different types of fractional partial differential equations to obtain exact solutions. The outcomes of this article are valuable to researchers and mathematicians who are interested in the solving fractional partial differential equations via integral transforms, which are considered as important techniques to handle these equations.

Complete two-loop QCD amplitudes for tW production at hadron colliders

We have calculated the complete two-loop QCD amplitudes for hadronic tW production by combining analytical and numerical techniques. The amplitudes have been first reduced to master integrals of eight planar and seven non-planar families, which can contain at most four massive propagators. Then a rational transformation of the master integrals is found to obtain a good basis so that the dimensional parameter decouples from the kinematic variables in the denominators of reduction coefficients. The master integrals are computed by solving their differential equations numerically. We find that the finite part of the two-loop squared amplitude is stable in the bulk of the phase space. After phase space integration and convolution with the parton distributions, it increases the LO cross section at the 13 TeV LHC by about 3%.