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Fractional differential equations (FDEs) involving a family of special functions and their solutions represent different physical phenomena. FDEs are characterizing and solving many problems of mathematical physics, chemistry, biology, and engineering. In this article, we establish an integral operator involving the family of incomplete H‐function (IHF) in its kernel. First, we derive the solutions for FDEs involving the generalized composite fractional derivative (GCFD) and integral operator associated with the incomplete H‐function. Several important special cases are revealed and analyzed. The main result derived in this study contains first‐order Volterra‐type integro‐differential equation describing the unsaturated nature of the free electron laser as a special case. Further, we give the graphical interpretation of the solution of FDEs. [ABSTRACT FROM AUTHOR]

In deep excavation engineering, the implementation of cut-off walls stands as a crucial measure to ensure structural support stability. However, the existing theories for dewatering design often overlook the variations in cut-off wall penetration depths, potentially compromising the efficacy of groundwater control strategies. Addressing this gap, this study conducts a comprehensive investigation into the dynamics of groundwater levels within confined aquifers during the dewatering process of strip excavation pits with cut-off walls. Central to this inquiry is the conceptualization of the entire excavation pit as a singular large-diameter well, with the open part beneath the cut-off wall in the confined aquifer serving as a constant-flux boundary. Employing an advanced analytical modeling approach, the study formulates a robust framework to describe the intricate interplay of unsteady groundwater flow phenomena. Leveraging the techniques of the Laplace and finite cosine transform methods, a semi-analytical solution is derived to elucidate groundwater drawdown patterns over time. The validation of the proposed solution against finite element method results underscores its fidelity and applicability. The parametric analysis reveals a dynamic evolution in drawdown characteristics within the confined aquifer, transitioning from initially cone-shaped distributions to more linear profiles that eventually stabilize with prolonged dewatering. This evolution is governed by the aquifer's inherent anisotropy and the barrier effect exerted by the depth of cut-off wall penetration. The parametric research also underscores the critical role of lateral boundary distance in influencing groundwater drawdown patterns. The presented solutions can be used to identify optimal penetration depth ratios tailored to specific parameters, thereby offering insights for optimizing dewatering strategies for deep excavation groundwater control. Moreover, a case study was included in this study using the proposed analytical solution, and a comparison was made with field data to validate the practical applicability of the approach. [ABSTRACT FROM AUTHOR]

More often, in the mathematical literature, the injectivity of the spherical Radon transform (SRT) for compactly supported functions is considered. In this article, an additional condition, for the reconstruction of an unknown function f∈C(R2)$f\in C(\mathbf {R}^2)$ (the support can be noncompact) using the circular Radon transform (CRT) over circles centered on a smooth simple curve is found. It is proved that this problem is equivalent to the injectivity of a so‐called two‐data CRT over circles centered on a smooth curve (can be a segment). Also, we present an inversion formula of the transform that uses the local data of the circular integrals to reconstruct the unknown function. Such inversions are the mathematical base of modern modalities of imaging, such as thermo‐ and photoacoustic tomography and radar imaging, and have theoretical significance. [ABSTRACT FROM AUTHOR]

In this study, we aim to investigate certain triple integral transform and its application to a class of partial differential equations. We discuss various properties of the new transform including inversion, linearity, existence, scaling and shifting, etc. Then, we derive several results enfolding partial derivatives and establish a multi-convolution theorem. Further, we apply the aforementioned transform to some classical functions and many types of partial differential equations involving heat equations, wave equations, Laplace equations, and Poisson equations as well. Moreover, we draw some figures to illustrate 3-D contour plots for exact solutions of some selected examples involving different values in their variables. [ABSTRACT FROM AUTHOR]

The goal of this effort is to determine the interaction among the heating and cooling processes in order to understand how solids behave when subjected to temperature changes. In this instance, the temperature, displacement, and stress relations are determined analytically and numerically while a thin annular disc is subjected to both the heating and cooling processes. The ability of a material to withstand stress is essential for the design of diverse mechanical structures that aim to enhance performance, durability, characteristics, and strength. This ability is demonstrated in many physical processes where the material structure crosses over into heating and cooling processes. Furthermore, memory derivatives used in the modelling of heat transfer equations more accurately depict the memory behaviour of an imagined disc and explain its physical significance.

Investigating solutions of differential equations has been an important issue for scientists. Researchers around the world have talked about different methods to solve differential equations. The type and order of the differential equation enable us to decide the method that we can choose to find the solution of the equation. One of these methods is the integral transform, which is the conversion of a real or complex valued function into another function by some algebraic operations. Integral transforms are used to solve many problems in mathematics and engineering, such as differential equations and integral equations. Therefore, new types of integral transforms have been defined, and existing integral transforms have been improved. One of the solution methods of many physical problems as well as initial and boundary value problems are integral transforms. Integral transforms were introduced in the first half of the 19th century. The first historically known integral transforms are Laplace and Fourier transforms. Over the time, other transforms that are used in many fields have emerged. The aim of this article is to describe the Mohand transform and to make applications of linear ordinary differential equations with constant coefficients without any major mathematical calculations This integral transform method is an alternative method to existing transforms such as Laplace transform and Kushare transform. When recent studies in the literature are examined, it can be said that Mohand transform is preferred because it is easy to apply.

This article addresses the two analytical methods, i.e., the new iterative transform method (NITM) and the homotopy perturbation transform method (HPTM), along with an Aboodh transform (AT), to approximate the nonlinear system of two-dimensional (2D) time-fractional Navier-Stokes (TFNS) equations. We take the time-fractional derivative in the form of Atangana-Baleanu (AB). The article's suggested examples examine the accuracy and efficacy of the proposed methods, while the graphs demonstrate their potential and effectiveness. The article also provides demonstrations of uniqueness and convergence. The aforementioned techniques are straightforward and support a high rate of convergence, which helps in understanding the dynamics of fractional nonlinear systems.

We study the non-linear transient gravity waves inside vast oceans with general topographies. These waves are generated following climate variations simulated by an external pressure acting on the ocean's surface. We use a perturbation method for the study. The present approach necessitates a mild slope of the topography. Quadratic solutions are obtained from nonlinear theory technique and illustrated. The reliability of the nonlinear (quadratic) solution is examined by a comparison between the trace of the bottom and the lowest streamline. The proposed model is shown to be strongly efficient in simulating the considered phenomenon, especially if the slope of the topography is not sharp. The features of the phenomenon under consideration are revealed and discussed mathematically and physically according to the nonlinear theory technique.

A mixed boundary-value problem for a poroelastic semi-infinite strip is formulated using Biot's theory. Two cases of boundary conditions are considered depending on the permeability of the longer sides. By using the integral transforms and matrix differential calculus, the original boundary value problem was reduced to a one-dimensional problem, which was then solved analytically. The explicit formulae are derived for the effective stress, pore pressure, and displacements. These functions are analyzed depending on the permeability of the boundary, the properties of the poroelastic material, and the compression profile along the sides. [ABSTRACT FROM AUTHOR]

The goal of this effort is to determine the interaction among the heating and cooling processes in order to understand how solids behave when subjected to temperature changes. In this instance, the temperature, displacement, and stress relations are determined analytically and numerically while a thin annular disc is subjected to both the heating and cooling processes. The ability of a material to withstand stress is essential for the design of diverse mechanical structures that aim to enhance performance, durability, characteristics, and strength. This ability is demonstrated in many physical processes where the material structure crosses over into heating and cooling processes. Furthermore, memory derivatives used in the modelling of heat transfer equations more accurately depict the memory behaviour of an imagined disc and explain its physical significance. [ABSTRACT FROM AUTHOR]

We study the non-linear transient gravity waves inside vast oceans with general topographies. These waves are generated following climate variations simulated by an external pressure acting on the ocean's surface. We use a perturbation method for the study. The present approach necessitates a mild slope of the topography. Quadratic solutions are obtained from nonlinear theory technique and illustrated. The reliability of the nonlinear (quadratic) solution is examined by a comparison between the trace of the bottom and the lowest streamline. The proposed model is shown to be strongly efficient in simulating the considered phenomenon, especially if the slope of the topography is not sharp. The features of the phenomenon under consideration are revealed and discussed mathematically and physically according to the nonlinear theory technique. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the monotonicity of the functions t \mapsto \frac {\sum _{k=0}^\infty a_k w_k(t)}{\sum _{k=0}^\infty b_k w_k(t)} and x \mapsto \frac {\int _\alpha ^\beta f(t) w(t,x) \mathrm {d} t}{\int _\alpha ^\beta g(t) w(t,x) \mathrm {d} t}, focusing on case where the monotonicity of a_k/b_k and f(t)/g(t) change once. The results presented also provide insights into the monotonicity of the ratios of two power series, two \mathcal {Z}-transforms, two discrete Laplace transforms, two discrete Mellin transforms, two Laplace transforms, and two Mellin transforms. Finally, we employ these monotonicity rules to present several applications in the realm of special functions and stochastic orders. [ABSTRACT FROM AUTHOR]

The exact solution of the poroelastic axisymmetric problem for a layer with a cylinderical hole is constructed under assumptions of Biot's model. The calculations provided by derived explicit formulas for full stress and pore pressure allow to state some important dependencies between the poroelastic stress state of the layer and type of poroelastic materials, loading types and height of the layer. [ABSTRACT FROM AUTHOR]

Distribution forecasts P over future quantities or events are routinely made in hydrology but usually traded for a (likelihood‐weighted) mean or median prediction to accommodate error measures or scoring functions such as the mean absolute error or mean squared error. Case in point is the so‐called KG efficiency (KGE) of Gupta et al. (2009, https://doi.org/10.1016/j.jhydrol.2009.08.003) and improvements thereof (Lamontagne et al., 2020, https://doi.org/10.1029/2020wr027101), which have rapidly gained popularity among hydrologists as alternative scoring functions to the commonly used Nash and Sutcliffe (1970, https://doi.org/10.1016/0022‐1694(70)90255‐6) efficiency, but are equally exclusive in how they quantify model performance using only single‐valued output of the quantities of interest. This point‐valued mapping necessarily implies a loss of information about model performance. This paper advocates the use of probabilistic watershed model training, evaluation and diagnostics. Distribution evaluation opens a mature literature on scoring rules whose strong statistical underpinning provides, as we will demonstrate, the theory, context and guidelines necessary for the development of robust information‐theoretically principled metrics for watershed signatures. These so‐called hydrograph functionals are scalar‐valued mappings of major behavioral watershed functions embodied in a strictly proper scoring rule. We discuss past developments that led to the current state‐of‐the‐art of distribution evaluation in hydrology and review scoring rules for dichotomous and categorical events, quantiles (intervals) and density forecasts. We are particularly concerned with elicitable functionals and scoring rule propriety, discuss the decomposition of scoring rules into a sharpness, reliability and entropy term and present diagnostically appealing strictly proper divergence scores of hydrograph functionals for flood frequency analysis, flow duration and recession curves. The usefulness and power of distribution‐based model evaluation and diagnostics by means of scoring rules is demonstrated on simple illustrative problems and discharge distributions simulated with watershed models using random sampling and Bayesian model averaging. The presented theory (a) enables a more complete evaluation of distribution forecasts, (b) offers a statistically principled means for watershed model training, evaluation, diagnostics and selection using hydrograph functionals and/or extreme events and (c) provides a universal framework for metric development of watershed signatures, promoting metric standardization and reproducibility. Plain Language Summary: The past decades have witnessed an unbridled growth in goodness‐of‐fit metrics of hydrologic models. These metrics may satisfy the needs of hydrologists but lack conforming theory and principles. This state of affairs (a) elicits improper model training and evaluation, (b) provokes and supports misguided inferences, (c) impedes statistically‐principled uncertainty quantification, metric standardization and development of universal model benchmarks and (d) obfuscates determination of whether the model has finished learning. What is more, most hydrologic model evaluation metrics in use today are rather exclusive in how they quantify model performance using only single‐valued simulated output of the quantities of interest. Predictive distributions derived from (quasi)‐Bayesian methods or ensembles are usually traded for a (likelihood‐weighted) mean or median prediction to accommodate error measures (scoring functions) such as the mean absolute error. This implies a large loss of information. This paper develops a distribution‐based approach to hydrologic model evaluation and diagnostics. Distribution evaluation opens the necessary theory and guidelines for development of robust information‐theoretically principled metrics of watershed signatures. These so‐called hydrograph functionals are scalar‐valued mappings of major behavioral watershed functions embodied in a strictly proper scoring rule. The hydrograph functionals offer a statistically principled means for hydrologic model evaluation, diagnostics and selection. Key Points: Scoring rules of hydrograph functionals provide an information‐theoretically principled means for watershed model training, evaluation, and diagnosticsWe present strictly proper (divergence) scores for flood frequency analysis, flow duration, and recession curvesPropriety and elicitability offer useful working paradigms for metric development of hydrograph functionals [ABSTRACT FROM AUTHOR]

This article addresses the two analytical methods, i.e., the new iterative transform method (NITM) and the homotopy perturbation transform method (HPTM), along with an Aboodh transform (AT), to approximate the nonlinear system of two-dimensional (2D) time-fractional Navier-Stokes (TFNS) equations. We take the time-fractional derivative in the form of Atangana-Baleanu (AB). The article's suggested examples examine the accuracy and efficacy of the proposed methods, while the graphs demonstrate their potential and effectiveness. The article also provides demonstrations of uniqueness and convergence. The aforementioned techniques are straightforward and support a high rate of convergence, which helps in understanding the dynamics of fractional nonlinear systems. [ABSTRACT FROM AUTHOR]

The problem of random oscillations generated by an internal defect within the neighborhood of the boundary of an elastic massive body at the prefailure stage is considered. The study is based on the results of the invariant method in the theory of acoustic emission (AE), according to which the statistical distribution of the values of the parameters of acoustic emission (AE) signals due to a defect obeys the stability condition when the body remains at the same prefailure stage. A mathematical model of a nonstationary wave field of displacements in an elastic massive body is constructed and the issues of the correctness of its application are studied. The problem is reduced to the study of a certain boundary integral equation in special classes of stochastic processes. We pose the problem of reconstructing and describing the nature of the random process of defect emission on the free boundary of the body based on AE signals. The data of numerical analysis are presented. [ABSTRACT FROM AUTHOR]

This study introduces a novel process identification method aimed at overcoming the challenge of accurately estimating process models when faced with deterministic disturbances, a common limitation in conventional identification methods. The proposed method tackles the difficult modeling problems due to deterministic disturbances by representing the disturbances as a linear combination of Laguerre polynomials and applies an integral transform with frequency weighting to estimate the process model in a numerically robust and stable manner. By utilizing a least squares approach for parameter estimation, it sidesteps the complexities inherent in iterative optimization processes, thereby ensuring heightened accuracy and robustness from a numerical analysis perspective. Comprehensive simulation results across various process types demonstrate the superior capability of the proposed method in accurately estimating the model parameters, even in the presence of significant deterministic disturbances. Moreover, it shows promising results in providing a reasonably accurate disturbance model despite structural disparities between the actual disturbance and the model. By improving the precision of process models under deterministic disturbances, the proposed method paves the way for developing refined and reliable control strategies, aligning with the evolving demands of modern industries and laying solid groundwork for future research aimed at broadening application across diverse industrial practices. [ABSTRACT FROM AUTHOR]

There are thirty one-parameter Laplace-type integral transforms, however, two-parameter Laplace-type integral transforms are few. Hence, the objective in the paper is to introduce and apply a new two-parameter Laplace-type integral transform, called the R-transform, to solve linear ordinary differential equations. Linear ordinary differential equations with constant coefficients were solved by taking the inverse R-transform. Also, two important theorems about the R-transform were proved. The theorems furnish us with the essential results for applications. Computations were manually done and confirmed with Mathematica software.

The objective of this paper is to explore the transient behaviors of fiber reinforced composites with two embedded collinear cracks subject to a sudden thermal shock. The orthotropic composites with both the vertically oriented and horizontally aligned fibers are examined to present a systematic comparison. By means of the integral transform methodology and singular integral equations, the thermal and elastic governing equations are solved numerically to determine the transient thermal process and stress intensifications around the crack tips. Parametric investigations are carried out for fiber volume fractions and crack distances. The findings would be beneficial to a more thorough knowledge of fibrous composites’ fracture characteristics under thermal shocks, which helps material design and structural optimization of the composites with fiber reinforcements.

As in the Laplace transform technique, a new integral technique called SEE (Sadiq-Emad-Eman) transform is submitted. Showed propositions and important properties of SEE integral technique for differentiate of functions and showed shifting property for SEE technique. Got transfer function of dynamic system in control theory applying SEE integral technique. Solved some important problems in control theory via SEE technique. Showed that the Laplace integral technique and every integral transform technique like the Laplace transform available in the literature are particular cases of the SEE technique.

This article utilizes the Aboodh residual power series and Aboodh transform iteration methods to address fractional nonlinear systems. Based on these techniques, a system is introduced to achieve approximate solutions of fractional nonlinear Korteweg-de Vries (KdV) equations and coupled Burger's equations with initial conditions, which are developed by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. As a result, the Aboodh residual power series and Aboodh transform iteration methods for integer-order partial differential equations may be easily used to generate explicit and numerical solutions to fractional partial differential equations. The results are determined as convergent series with easily computable components. The results of applying this process to the analyzed examples demonstrate that the new technique is very accurate and efficient. [ABSTRACT FROM AUTHOR]

This research article introduces the four-dimensional natural transform Adomian decomposition method (FNADM) for solving the (3+1)-dimensional time-singular fractional coupled Burgers' equation, along with its associated initial conditions. The FNADM approach represents a fusion of four-dimensional natural transform techniques and Adomian decomposition methodologies. In order to observe the influence of time-Caputo fractional derivatives on the outcomes of the aforementioned models, two examples are illustrated along with their three-dimensional figures. The effectiveness and reliability of this approach are validated through the analysis of these examples related to the (3+1)-dimensional time-singular fractional coupled Burgers' equations. This study underscores the method's applicability and effectiveness in addressing the complex mathematical models encountered in various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

In this paper, we derive a new generalized integral transform on Dzherbashian-Nersesian fractional operator and give some special cases. We make a generalization of the application of integral transformations to different fractional operators, where several previous results can be invoked from a single relation. We also use the new results obtained to solve some fractional differential equations involving the recent revival of Dzherbashian-Nersesian fractional operators. [ABSTRACT FROM AUTHOR]

The problem of complete characterization of injectivity sets of the circular Radon transform on the space of compactly supported continuous functions is solved only for two-dimensional case. In particular, any small curve in the plane which is not contained in a straight line is an injectivity set. The problem becomes more complicated without compactness of support condition. The problem arises to find an additional condition that allows one to reconstruct an unknown function f ∈ C (R 2) (not necessarily with compact support) using the circular Radon transform (CRT) with detectors placed on an arc. In this article, an additional condition is found. This problem is equivalent to the injectivity of a transform that consists of CRT and weighted CRT. Also, in this article, a new iterative inversion formula for the transform is presented. This formula allows the reconstruction of an unknown function using the local data of the circular integrals of the function over circles centered on an arc. Such inversions are of theoretical importance in many areas of mathematics and are required in the mathematical models of thermo and photoacoustic tomography, radar imaging, and others. [ABSTRACT FROM AUTHOR]

The novelty of the presented results is the deriving of exact solutions of axisymmetric poroelasticity problems for finite solid and hollow cylinders. The statement of these contact problems is given under the assumptions of Biot's model for poroelastic materials. A simulation of the poroelastic response of a cylinder is implemented using the apparatus of boundary problems. By the method of integral transforms, the original boundary problem is reduced to a one-dimensional boundary problem in the transform domain. Based on the exact solution of the latter one, it is manageable to derive the explicit formulas for displacements, stress, and pore pressure. The investigation of contact stress and pore pressure distribution inside the cylinder regarding its geometrical ratios, type of applied loading, and poroelastic material is carried out. [ABSTRACT FROM AUTHOR]

This work introduces the double Laplace formable transform (DLFT), a novel double transform. A number of fundamental features and functions have been demonstrated using the newly defined double transform definition. Further, the existence, convolution, and derivative theorems of the DLFT are given and proven. Afterward, a few families of partial differential equations (PDEs) are solved using these results. There are new methods for solving PDE families. The latter solves the advection-diffusion equation, the Klein-Gordon equation, the telegraph equation, and other well-known PDEs precisely. According to the results, the DLFT is a useful and effective strategy for dealing with this kind of issue. [ABSTRACT FROM AUTHOR]

The blow-up phenomenon may occur in nonlinear integral equations and differential equations, which has important significance for the simulation of some practical problems. This paper is devoted to predicting the blow-up time for convolution Volterra-Hammerstein integral equations using the series expansion of the solution about the origin. First, the finite-term series expansion for the solution about the origin is obtained via Picard iteration and symbolic computation, which has high accuracy near the origin. Second, the Padé approximation of the series expansion is made to extend the convergence region of the series. The blow-up time of the solution is estimated by calculating the smallest positive root of the denominator of the rational function. Third, an integral transform is performed to the series expansion to convert the branch point to a pole of order one at the blow-up time, so the predicted accuracy of the blow-up time is improved remarkably. Numerical examples illustrate that the proposed method is very efficient for estimating the blow-up time of Volterra integral equations and some kinds of differential equations. [ABSTRACT FROM AUTHOR]

We extend previous research to derive three additional M-1-dimensional integral representations over the interval [ 0 , 1 ] . The prior version covered the interval [ 0 , ∞ ] . This extension applies to products of M Slater orbitals, since they (and wave functions derived from them) appear in quantum transition amplitudes. It enables the magnitudes of coordinate vector differences (square roots of polynomials) | x 1 − x 2 | = x 1 2 − 2 x 1 x 2 cos θ + x 2 2 to be shifted from disjoint products of functions into a single quadratic form, allowing for the completion of its square. The M-1-dimensional integral representations of M Slater orbitals that both this extension and the prior version introduce provide alternatives to Fourier transforms and are much more compact. The latter introduce a 3M-dimensional momentum integral for M products of Slater orbitals (in M separate denominators), followed in many cases by another set of M-1-dimensional integral representations to combine those denominators into one denominator having a single (momentum) quadratic form. The current and prior methods are also slightly more compact than Gaussian transforms that introduce an M-dimensional integral for products of M Slater orbitals while simultaneously moving them into a single (spatial) quadratic form in a common exponential. One may also use addition theorems for extracting the angular variables or even direct integration at times. Each method has its strengths and weaknesses. We found that these M-1-dimensional integral representations over the interval [ 0 , 1 ] are numerically stable, as was the prior version, having integrals running over the interval [ 0 , ∞ ] , and one does not need to test for a sufficiently large upper integration limit as one does for the latter approach. For analytical reductions of integrals arising from any of the three, however, there is the possible drawback for large M of there being fewer tabled integrals over [ 0 , 1 ] than over [ 0 , ∞ ] . In particular, the results of both prior and current representations have integration variables residing within square roots asarguments of Macdonald functions. In a number of cases, these can be converted to Meijer G-functions whose arguments have the form (a x 2 + b x + c) / x , for which a single tabled integral exists for the integrals from running over the interval [ 0 , ∞ ] of the prior paper, and from which other forms can be found using the techniques given therein. This is not so for integral representations over the interval [ 0 , 1 ] . Finally, we introduce a fourth integral representation that is not easily generalizable to large M but may well provide a bridge for finding the requisite integrals for such Meijer G-functions over [ 0 , 1 ] . [ABSTRACT FROM AUTHOR]

Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential equations with constant coefficients, utilizing the Kamal transform, where we focus on first- and second-order linear equations. In particular, in this work, we employ the Kamal transform to determine the Hyers–Ulam stability and Hyers–Ulam stability constants for first-order complex constant coefficient differential equations and, for second-order real constant coefficient differential equations, improving previous results obtained by using the Kamal transform. In a section of examples, we compare and contrast our results favorably with those established in the literature using means other than the Kamal transform. [ABSTRACT FROM AUTHOR]

The study focuses on thermoelastic behavior of the time-fractional heat conduction equation in a thick hollow cylinder by quasi-static approach. The utilization of the Caputo fractional derivative has been employed. The upper and lower surfaces of the cylinder have been assigned arbitrary temperature values, while the boundaries of the cylinder remain at a constant temperature of zero. The integral transform technique has been utilized to solve the conduction equation. A mathematical model has been prepared for Copper material. The fractional-order thermoelasticity for distinct fractional values of time variable has been discussed and the results are presented graphically. [ABSTRACT FROM AUTHOR]

AbstractThe main objective of this work is to introduce a novel generalization of double transformations called the triple Laplace–ARA–Sumudu transform (TLARAST). This hybrid transformation extends the concepts of double Laplace–Sumudu, double Laplace–ARA and double ARA–Sumudu transforms into a triple hybrid transform. The article provides the definition of TLARAST and investigates its fundamental properties, including existence, inverses and related theorems. Furthermore, new results concerning TLARAST for partial derivatives and the theorem of multi-convolution are introduced and discussed. The practicality and efficacy of TLARAST are demonstrated by applying it to solve various types of partial differential equations with significant applications in physics and other scientific fields, such as the heat equation, Laplace equation, Poisson equation and wave equation. The solutions are illustrated through figures created using Mathematica software. Overall, this study underscores the usefulness and efficiency of TLARAST in solving partial differential equations involving multiple variables.

In deep excavation engineering, the implementation of cut-off walls stands as a crucial measure to ensure structural support stability. However, the existing theories for dewatering design often overlook the variations in cut-off wall penetration depths, potentially compromising the efficacy of groundwater control strategies. Addressing this gap, this study conducts a comprehensive investigation into the dynamics of groundwater levels within confined aquifers during the dewatering process of strip excavation pits with cut-off walls. Central to this inquiry is the conceptualization of the entire excavation pit as a singular large-diameter well, with the open part beneath the cut-off wall in the confined aquifer serving as a constant-flux boundary. Employing an advanced analytical modeling approach, the study formulates a robust framework to describe the intricate interplay of unsteady groundwater flow phenomena. Leveraging the techniques of the Laplace and finite cosine transform methods, a semi-analytical solution is derived to elucidate groundwater drawdown patterns over time. The validation of the proposed solution against finite element method results underscores its fidelity and applicability. The parametric analysis reveals a dynamic evolution in drawdown characteristics within the confined aquifer, transitioning from initially cone-shaped distributions to more linear profiles that eventually stabilize with prolonged dewatering. This evolution is governed by the aquifer’s inherent anisotropy and the barrier effect exerted by the depth of cut-off wall penetration. The parametric research also underscores the critical role of lateral boundary distance in influencing groundwater drawdown patterns. The presented solutions can be used to identify optimal penetration depth ratios tailored to specific parameters, thereby offering insights for optimizing dewatering strategies for deep excavation groundwater control. Moreover, a case study was included in this study using the proposed analytical solution, and a comparison was made with field data to validate the practical applicability of the approach.

Investigating solutions of differential equations has been an important issue for scientists. Researchers around the world have talked about different methods to solve differential equations. The type and order of the differential equation enable us to decide the method that we can choose to find the solution of the equation. One of these methods is the integral transform, which is the conversion of a real or complex valued function into another function by some algebraic operations. Integral transforms are used to solve many problems in mathematics and engineering, such as differential equations and integral equations. Therefore, new types of integral transforms have been defined, and existing integral transforms have been improved. One of the solution methods of many physical problems as well as initial and boundary value problems are integral transforms. Integral transforms were introduced in the first half of the 19th century. The first historically known integral transforms are Laplace and Fourier transforms. Over the time, other transforms that are used in many fields have emerged. The aim of this article is to describe the Mohand transform and to make applications of linear ordinary differential equations with constant coefficients without any major mathematical calculations This integral transform method is an alternative method to existing transforms such as Laplace transform and Kushare transform. When recent studies in the literature are examined, it can be said that Mohand transform is preferred because it is easy to apply. [ABSTRACT FROM AUTHOR]

With the rapid development of of fuzzy Integral equations, the problem of finding its solution becomes increasingly important for knowledge propagation and putting researchers wisdom to work. A recent development trend of fuzzy integral equations is modeling many problems in the field of science and technology. If the unknown function in the considered equation has a solution in terms of infinite series expansion, this proposed method becomes more accurate to find the exact solution. According on the parametric form of a fuzzy number, a fuzzy integral equation converts to two systems of integral equations in the crisp case, and then proposed method is achieved to obtain the exact fuzzy solutions of fuzzy Volterra integral equations. To validate the effectiveness of proposed method, some examples of considered equations are solved and the results show that it achieved the best performance. [ABSTRACT FROM AUTHOR]

This article discusses a three-dimensional analog of the zero-order Hankel transform, which is an integral transform used in the study of coherent states. The transform acts on functions of three variables and has similar properties to the usual Hankel transform. The article presents key properties of the transform, such as orthogonality, Plancherel's theorem, inversion formula, composition with differential operators, similarity formula, and convolution. The three-dimensional analog of the Hankel transform has applications in the theory of partial differential equations. The article also provides notation for vector operations and defines the spaces and used in the analysis. The author acknowledges the contributions of others and declares no conflicts of interest. [Extracted from the article]

We propose an integral transform, called metamorphism, which allows us to reduce the order of a differential equation. For example, the second-order Helmholtz equation is transformed into a first-order equation, which can be solved by the method of characteristics. [ABSTRACT FROM AUTHOR]

This study embarks on an exploration of the thermal deflection characteristics of finite hollow cylinders, employing the space-time fractional heat conduction equation within a quasi-static framework. Heat application is executed on the upper surface of the cylinder, whilst maintaining a zero-temperature condition on the remaining boundaries. Temperature distribution across the cylinder is determined using the integral transform technique, a method ensuring precision in the computation of thermal responses. The discourse on thermal deflection is grounded in the principles of fractional diffusion wave theory, a contemporary approach providing deeper insights into heat conduction dynamics. Numerical analyses are presented, illustrating transient and long-range interaction responses of the hollow cylinder under various diffusion scenarios, namely sub-diffusion, normal diffusion, and super-diffusion. [ABSTRACT FROM AUTHOR]

The main objective of this work is to introduce a novel generalization of double transformations called the triple Laplace–ARA–Sumudu transform (TLARAST). This hybrid transformation extends the concepts of double Laplace–Sumudu, double Laplace–ARA and double ARA–Sumudu transforms into a triple hybrid transform. The article provides the definition of TLARAST and investigates its fundamental properties, including existence, inverses and related theorems. Furthermore, new results concerning TLARAST for partial derivatives and the theorem of multi-convolution are introduced and discussed. The practicality and efficacy of TLARAST are demonstrated by applying it to solve various types of partial differential equations with significant applications in physics and other scientific fields, such as the heat equation, Laplace equation, Poisson equation and wave equation. The solutions are illustrated through figures created using Mathematica software. Overall, this study underscores the usefulness and efficiency of TLARAST in solving partial differential equations involving multiple variables. [ABSTRACT FROM AUTHOR]

In this paper, the inverse problem of determining the time‐dependent reaction–diffusion coefficient in the Cauchy problem for the time‐fractional diffusion equation with the Hilfer operator by a single observation at the point x=0$$ x=0 $$ of the diffusion process is studied. To represent the solution of the direct problem, the fundamental solution of this equation is constructed and properties of this solution are investigated. The fundamental solution contains Fox's H$$ H $$‐functions widely used in fractional calculus. The fundamental solution for the fractional diffusion operator with the Hilfer derivative is also obtained in terms of the Yang Y$$ Y $$‐function, and it is shown that they coincide. Using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown coefficient, which will be used in investigation of inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation, the contracted mapping principle is applied. The local existence and global uniqueness results and the conditional stability estimate of solution are proven. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This article utilizes the Aboodh residual power series and Aboodh transform iteration methods to address fractional nonlinear systems. Based on these techniques, a system is introduced to achieve approximate solutions of fractional nonlinear Korteweg-de Vries (KdV) equations and coupled Burger’s equations with initial conditions, which are developed by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. As a result, the Aboodh residual power series and Aboodh transform iteration methods for integer-order partial differential equations may be easily used to generate explicit and numerical solutions to fractional partial differential equations. The results are determined as convergent series with easily computable components. The results of applying this process to the analyzed examples demonstrate that the new technique is very accurate and efficient.

In this paper, we derive a new generalized integral transform on Dzherbashian–Nersesian fractional operator and give some special cases. We make a generalization of the application of integral transformations to different fractional operators, where several previous results can be invoked from a single relation. We also use the new results obtained to solve some fractional differential equations involving the recent revival of Dzherbashian-Nersesian fractional operators.

An internal heat source is assumed to act on a cylindrical body with radiation-like boundary conditions to explore the memory-dependent thermoelastic response of a solid object. The top and bottom surfaces of the solid cylinder are subjected to additional heating conditions. To obtain the thermal behaviour of the considered medium, the integral transform method is used, while the inversion solution of the heat transfer equation, the thermoelastic displacement and stress functions are presented in the Laplace domain due to the complexity of the calculation. To understand the numerical calculations, the material properties of aluminium metal are taken into account, and all the obtained results are presented graphically.

The proposed study is focused to introduce a novel integral transform op-erator, called Generalized Bivariate (GB) transform. The proposed trans-form includes the features of the recently introduced Shehu transform, ARA transform, and Formable transform. It expands the repertoire of existing Laplace-type bivariate transforms. The primary focus of the present work is to elaborate fashionable properties and convolution theorems for the proposed transform operator. The existence, inversion, and duality of the proposed transform have been established with other existing transforms. Implementation of the proposed transform has been demonstrated by ap-plying it to different types of differential and integral equations. It validates the potential and trustworthiness of the GB transform as a mathematical tool. Furthermore, weighted norm inequalities for integral convolutions have been constructed for the proposed transform operator.