Analytical Series Solution Research Articles

Computational Approaches to Solving Some Partial Differential Equations and Neutrosophic Partial Differential with Variable Coefficients Using the Laplace Residual Power Series Method

We employ the Laplace Residual Power Series Method to approximate analytical solutions for differential equations and neutrosophic differential equations with associated parameters, including non-homogeneous equations and fractional formulas in partial differential equations (PDEs). This approach showcases the method's simplicity, effectiveness, and robustness in deriving analytical series solutions for PDEs that involve associated parameters, especially in the context of fractional differential equations. Several practical uses of LRPSM with an emphasis on non-homogeneous and partial differential equations and neutrosophic equations with fractions (PDEs). These applications are significant in a variety of scientific and engineering domains that simulate complicated dynamic system such as anomalous diffusion in physics, viscoelastic material modeling in engineering and signal processing.

Diffusional penetration of aerosols through a tube: Comparison between Monte Carlo simulation of Brownian particle trajectory and the numerical solution of the advection–diffusion equation

Penetration of nanometer-sized, diffusive aerosol particles through a circular tube has been determined by two numerical methods. One method consisted in the simulation of the trajectories of Brownian particles suspended in a flowing fluid medium. The other was the numerical solution of the advection–diffusion equation. For any given value of the particle diffusion coefficient, penetration, i.e. the fraction of particles that avoid diffusion loss to the wall and exit the tube, calculated by the two methods agreed fairly well with each other for the three types of fluid flow tested (uniform, developing, and fully developed parabolic flows). For the case of parabolic flow there exists an analytical series solution which has been successfully compared with experimental results in a relatively large number of past investigations. The results obtained by the two numerical methods have also shown an excellent agreement with this analytical solution. The Brownian dynamics simulation method requires a larger computer time, but its simplicity allows examination of other aerosol flow processes too difficult to study either experimentally or by means of conventional differential equations. Aerosol penetration in transient, developing flow has never been addressed before, neither experimentally nor theoretically. The results reported in this paper are the first ones.

Direct and some inverse problems for a generalized diffusion equation with variable coefficients

Direct and two inverse problems for a Legendre equation involving integral convolution in time are studied. The inverse problems are ill-posed in the sense of Hadamard. The analytical series solutions of the problems are constructed by using method of variable separation. The determination of only u(x, y) is studied in the direct problem, the recovery of a pair of functions, i.e., {u(x,y),f(x)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{u(x,y), f(x)\\}$$\\end{document} with appropriate addition data at some T is investigated in the 1st inverse problem while the identification of a pair of functions, i.e., {u(x,y),q(y)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{u(x,y), q(y)\\}$$\\end{document} with an integral type data is considered in the 2nd inverse problem. By imposing certain regularity conditions, the unique existence of series solutions is developed. We provided some numerical examples to illustrate our results for the inverse problems.

Melting heat transfer of a quadratic stratified Jeffrey nanofluid flow with inclined magnetic field and thermophoresis

Recently, researchers are facing great challenges to form the homogeneous solutions of various liquids at biomedical and industrial levels. Such homogeneous solutions can be controlled through stratification phenomenon directly which is ignored by the researchers in the existing literature specifically quadratic stratification and melting heat mechanism which is valid only for higher temperature processes. Thus, to form the homogeneous and stable liquid solutions for various industrial processes, we investigate the melting heat phenomenon in quadratic stratified Jeffery Nano fluid flow deformed due to linearly stretching sheet along with stagnation point. Inclined constant magnetic field is implemented on the fluid through an arbitrary angel. Characteristics of heat and transportations are disclosed through viscous dissipation, Brownian motion and thermphoresis. Temperature and concentration of the surrounding fluids are assumed higher than the stretching surface. The resultant governing equations are reduced to ordinary differential equations through proper transformations. Convergent analytical series solutions are computed via homotopic approach. Impact of various emerging parameters on temperature, velocity and concentration fields are illustrated and discussed comprehensively. Sherwood and Nusselt numbers and drag force are scrutinized mathematically, physically and graphically. It is analyzed from the study that (i) dominant melting reduces the temperature field (ii) higher thermal and solutal stratification decay the temperature and concentration fields respectively (iii) higher thermophoresis enhances the temperature field. Further, it is concluded that stable and homogeneous solutions may be made by increasing melting and reducing stratification phenomena. This study will help us to provide actual amount of heat for heating processes in industries and biomedicine which is the basic requirement for excellent quality of products.

Applying Conformable Double Sumudu – Elzaki Approach to Solve Nonlinear Fractional Problems

In this study, a conformable double Sumudu-Elzaki (CDSE) transformation and a decomposition method are combined to develop a new method that solves nonlinear problems considering some specific conditions. This combination can be referred to as the CDSE-Decomposition method. Furthermore, we explain and discuss the main features and main results of the presented method. The CDSE-Decomposition method presents analytic series solutions with high convergence to the exact solution in a closed form. The benefit of utilizing the proposed technique is that it presents analytic series solutions to the objective equations without the requirement for any constrained assumptions, transformation or discretization. In addition, various numerical experiments are presented to prove the efficiency of the obtained method. The results show the power and effectiveness of the proposed approach in handling a range of physical and engineering problems that arize in mathematics.

A new approach in handling one-dimensional time-fractional Schrödinger equations

<abstract> <p>Our aim of this paper was to present the accurate analytical approximate series solutions to the time-fractional Schrödinger equations via the Caputo fractional operator using the Laplace residual power series technique. Furthermore, three important and interesting applications were given, tested, and compared with four well-known methods (Adomian decomposition, homotopy perturbation, homotopy analysis, and variational iteration methods) to show that the proposed technique was simple, accurate, efficient, and applicable. When there was a pattern between the terms of the series, we could obtain the exact solutions; otherwise, we provided the approximate series solutions. Finally, graphical results were presented and analyzed. Mathematica software was used to calculate numerical and symbolic quantities.</p> </abstract>

Analytical Approximate Solutions of Caputo Fractional KdV‐Burgers Equations Using Laplace Residual Power Series Technique

The KdV‐Burgers equation is one of the most important partial differential equations, established by Korteweg and de Vries to describe the behavior of nonlinear waves and many physical phenomena. In this paper, we reformulate this problem in the sense of Caputo fractional derivative, whose physical meanings, in this case, are very evident by describing the whole time domain of physical processing. The main aim of this work is to present the analytical approximate series for the nonlinear Caputo fractional KdV‐Burgers equation by applying the Laplace residual power series method. The main tools of this method are the Laplace transform, Laurent series, and residual function. Moreover, four attractive and satisfying applications are given and solved to elucidate the mechanism of our proposed method. The analytical approximate series solution via this sweet technique shows excellent agreement with the solution obtained from other methods in simple and understandable steps. Finally, graphical and numerical comparison results at different values of α are provided with residual and relative errors to illustrate the behaviors of the approximate results and the effectiveness of the proposed method.

LRPS Method for Solving Linear Partial Differential Equations and Neutrosophic Differential Equations of Fractional Order with Numerical Solutions

In this work, fractional partial equations' and neutrosophic fractional partial equations analytical series solutions are presented, we consider the fractional derivative in the meaning of Caputo in these formulas. We offer a novel objective method the LRPS which is a strong instrument for precise analytically and numerical solutions to these problems by setting an excellent example, we stress precision, effectiveness, and application style, also we can find exact answers when there is a pattern between the series' parts; alternatively, we can only offer approximations. The Mathematica application is used to assess the numerical and graphical findings to make sure the solutions generated are accurate and that the approach can be modified to solve this kind of this problem. The findings obtained demonstrated that our current procedure is appropriate and efficient for resolving PDEs.

A Novel Approach for Solving Nonlinear Time Fractional Fisher Partial Differential Equations

This study focuses on solving non-linear time fractional Fisher partial differential equations using analytical series solutions. The authors consider the Caputo fractional derivative in their formulas, which adds accuracy to the results. They introduce a novel method called LRPS which proves to be a powerful tool for obtaining precise analytical and numerical solutions for these equations. The LRPS method emphasizes precision, effectiveness, and practical application, making it suitable for various fields such as physics, engineering, and finance. Due to the importance of accuracy, effectiveness and method of application in this approach, it is highlighted that accurate solutions can be obtained when there is a pattern in the parts of the series, while approximate estimates are provided otherwise. The LRPS method is presented as a powerful technique specifically designed for solving nonlinear fractional Fisher partial differential equations.

Vibration isolation of infinitely periodic pile barriers for anti-plane shear waves: An exact series solution

The application of periodic pile barriers for isolating ambient vibrations has emerged as a significant research topic, which is essentially the problem of wave scattering by periodically distributed structures. This study presents an analytical series solution for anti-plane shear (SH) wave scattering by infinitely periodic pile barriers using the wave function expansion method, incorporating the wave field characteristics of infinitely periodic distributed scatterers subjected to plane waves. From the perspective of periodic theory, the wave fields around each scatterer are completely identical, differing only by a phase shift (frequency domain) or a time difference (time domain). As a result, as long as the boundary condition of one scatterer is satisfied, boundary conditions of the remaining scatterers can be satisfied simultaneously. This novel approach allows the selection of an arbitrary single pile as a reference in the infinitely periodic model to be solved, yielding accurate results without truncation errors and significantly reducing computational and storage requirements. A FORTRAN code is developed to achieve numerical evaluation of the theoretical formulation, followed by verification of the solution's accuracy using two degenerate examples. Numerical examples are conducted, focusing on the influence of pile stiffness, pile spacing, and pile type on the vibration isolation. The results demonstrate that decreasing the spacing between piles can effectively increase the bandgap width and reduce the displacement response. The screening effectiveness of solid piles is better than that of hollow pipe piles and filled pipe piles. In vibration mitigation design for engineering projects, the larger the shear stiffness ratio of the pile-soil is not always better, an economical and reasonable pile stiffness should be selected.

Homotopy analysis method and its convergence analysis for a nonlinear simultaneous aggregation-fragmentation model

In this article, we focus on addressing the missing convergence analysis of the homotopy analysis method (HAM) for solving pure aggregation and pure fragmentation population balance equations [Kaur et al. (2023), J. Math. Anal. Appl., 512(2), 126166]. This technique is further extended to determine analytical series solutions for a simultaneous aggregation–fragmentation (SAF) population balance equation. The convergence analysis of the extended approach for a SAF equation is performed using the concept of contraction mapping in the Banach space. The HAM method enables us to derive recursive formulas to obtain series solutions, distinguishing it from traditional numerical approaches. One noteworthy advantage of HAM is its capability to solve both linear and nonlinear differential equations without resorting to discretization, while incorporating a convergence control parameter. Given the complex nature of the SAF equation, only a single analytical solution has been available, specifically for a constant aggregation kernel and a binary breakage kernel with a linear selection function. However, our study presents new series solutions for the number density functions, considering the combination of sum and product aggregation kernels with binary breakage kernels and linear/quadratic selection functions. These particular solutions have not been previously documented in the existing literature. To verify the accuracy and efficiency of the proposed approach, the results with the finite volume scheme [Singh et al. (2021), J. Comput. Phys., 435, 110215] for establishing the accuracy and effectiveness of the proposed approach.

Levy-type analytical series solution for the three-dimensional free vibrations of functionally graded material rectangular plates with piezoelectric layers

Analytical solutions founded on three-dimensional theories play a crucial role in evaluating the credibility and precision of different plate theories and numerical methodologies. While Levy-type analytical solutions are widely recognized, they have been primarily confined to purely elastic plates. This study introduces a Levy-type analytical series solution for three-dimensional vibrations in a sandwich rectangular plate featuring a functionally graded material (FGM) core, along with piezoelectric material (PM) layers on the top and bottom surfaces. The behaviors of the FGM and PM layers were described using three-dimensional elasticity and piezoelasticity theories, respectively. In this study, the displacement functions and electric potential of each layer were expanded by Fourier series and polynomial auxiliary functions. An analytical series solution was then established by satisfying the governing equations of each layer, the mechanical and electric boundary conditions on the six faces of the plate, and the continuity conditions on the interfaces between the PM and FGM layers. To validate the proposed solutions, in-depth convergence studies were conducted for the vibration frequencies of the first six modes of sandwich square plates with various boundary conditions on the other pair of side faces. The well-converged results were then compared with published data based on various plate theories to verify the accuracy of these published data. Finally, accurate nondimensional frequencies were tabulated for the first six modes of sandwich rectangular plates with various aspect ratios, thickness-to-width ratios, PM-to-FGM layer thickness ratios, power law indices for the FGM layer, and six combinations of boundary conditions. These new numerical results when piezoelectric coupling is considered should be very useful to future analytical and numerical studies.

Adapting partial differential equations via the modified double ARA-Sumudu decomposition method

In order to get analytical series solutions of partial differential equations while taking certain conditions into consideration, this paper presents the new procedure of using the new double ARA-Sumudu transform (DARA-ST) in conjunction with the modified decomposition method in handling nonlinear equations. The solutions of the nonlinear terms in these equations are obtained depending on defining series of iterations. The suggested method has the merits of producing accurate answers and it is simply analytically applicable to the provided problems. Moreover, fundamental theorems and properties of DARA-ST are proposed. Numerical examples are provided to show how this strategy might be used and how beneficial it is. The obtained results demonstrate the method’s potential for resolving further nonlinear problems and getting exact solutions.

Heat transfer in MHD thin film flow with concentration using lie point symmetry approach

Heat and mass transfer in a viscous film on an unsteady stretching surface in the presence of a variable magnetic field is investigated using Lie symmetry analysis. Combination of translational and scaling Lie point symmetries is used to obtain invariants of the model. The deduced invariants provide a new generalized class of similarity transformations that have not been used before. These transformations restructure the governing problem in a system of eight-parameter nonlinear ordinary differential equations. Analytic series solutions are obtained for the resulting system of equations for different values of these parameters and are illustrated graphically. We found that coefficients of the translational symmetries do not have any effect on the solutions however, coefficients of the scaling symmetries significantly affect the variation of temperature and concentration distribution across the fluid film and help in controlling the heat and mass transfer rates. Also, increasing the value of unsteadiness and magnetic parameter decreases the film thickness and promotes a uniform temperature and concentration across film thickness. Furthermore, we found that at the lower values of Prandtl and Schmidt number, diffusion dominates the heat and mass transfer while at the higher values advection dominates the heat and mass transfer.

GKZ hypergeometric systems of the three-loop vacuum Feynman integrals

We present the Gel’fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems of the Feynman integrals of the three-loop vacuum diagrams with arbitrary masses, basing on Mellin-Barnes representations and Miller’s transformation. The codimension of derived GKZ hypergeometric systems equals the number of independent dimensionless ratios among the virtual masses squared. Through GKZ hypergeometric systems, the analytical hypergeometric series solutions can be obtained in neighborhoods of origin including infinity. The linear independent hypergeometric series solutions whose convergent regions have non-empty intersection can constitute a fundamental solution system in a proper subset of the whole parameter space. The analytical expression of the vacuum integral can be formulated as a linear combination of the corresponding fundamental solution system in certain convergent region.

Mathematical modelling of unsteady Oldroyd-B fluid flow due to stretchable cylindrical surface with energy transport

The unsteady boundary layer flow and thermal transportation analysis of an incompressible Oldroyd-B nanofluid driven by stretching cylinder is scrutinized with analytically approach in this study. The governing unsteady boundary layer equations (PDEs) are first established, and then these PDEs are converted into highly nonlinear ordinary differential equations (ODEs) using appropriate similarity transformations. The obtained equations are then solved in the form of series solutions via homotopy analysis method (HAM) for the velocity, temperature, and concentration fields. Additionally, the convergence analysis of the analytic series solutions is discussed. The model under consideration predicts the features of both the relaxation and retardation times effects. The aspects of thermophoresis and Brownian motion characteristics due to nanoparticles are studied by employing Buongiorno's model. The findings demonstrated that when the curvature parameter increases, the velocity, temperature, and concentration distributions near the cylinder's surface enhanced. Moreover, the velocity profile, for relaxation and the retardation times parameters, showed the opposite behavior. Also, the unsteadiness parameter enhanced the fluid velocity. The comparison of some current and old results is remarkable.

A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution

In this paper, a new method based on an accurate analytical series solution for free vibration of triangular isotropic and orthotropic plates is presented. The proposed solution is expressed in terms of undetermined arbitrary coefficients, which are exactly satisfied by the governing differential equation in free vibration. The approach used is based on an innovative extension of the superposition method through the application of a modified system of trigonometric functions. The boundary conditions for bending displacements and bending rotations on the sides of the triangular plate led to an infinite system of linear algebraic equations in terms of the undetermined coefficients. Following this development, the paper then presents an algorithm to solve the boundary value problem for isotropic and orthotropic triangular plates for any kinematic boundary conditions. Of course, the boundary conditions with zero displacements and zero rotations on all sides correspond to the case when the plate is fully clamped all around. The convergence of the proposed method is examined by numerical simulation applying stringent accuracy requirements to fulfill the prescribed boundary conditions. Some of the computed numerical results are compared with published results and finally, the paper draws significant conclusions.

Dynamics of heat transport in flow of non‐linear Oldroyd‐B fluid subject to non‐Fourier's theory

AbstractTo study non‐Fourier heat and mass transport in the stagnation point flow of magnetized Oldroyd‐B fluid due to stretching cylinder, the Cattaneo–Christov heat flux model is used in this investigation. Further, as the controlling agents for thermal and solutal transport in the fluid flow, the heat generation/absorption source and chemical reaction are also considered. The formulations of a such physical phenomenon are going to form the PDEs. Through appropriate similarity variables, these governing partial differential equations for flow and energy transport are converted into the ordinary differential. The analytical series solutions are obtained through the use of a homotopic approach. The graphical upshots are conducted for velocity fields, temperature, and concentration distributions. In addition, energy transport analysis is performed for two kinds of surface heating mechanisms, namely the prescribed surface temperature (PST) and constant wall temperature (CWT). The outcomes of the current investigation revealed that a higher rate of heat transfer is observed in the case of CWT as compared to PST. Moreover, the increasing values of thermal and solutal relaxation time parameters reduce the heat and mass transport in the fluid flow, respectively.

Direct Power Series Approach for Solving Nonlinear Initial Value Problems

In this research, a new approach for solving fractional initial value problems is presented. The main goal of this study focuses on establishing direct formulas to find the coefficients of approximate series solutions of target problems. The new method provides analytical series solutions for both fractional and ordinary differential equations or systems directly, without complicated computations. To show the reliability and efficiency of the presented technique, interesting examples of systems and fractional linear and nonlinear differential equations of ordinary and fractional orders are presented and solved directly by the new approach. This new method is faster and better than other analytical methods in establishing many terms of analytic solutions. The main motivation of this work is to introduce general new formulas that express the series solutions of some types of differential equations in a simple way and with less calculations compared to other numerical power series methods, that is, there is no need for differentiation, discretization, or taking limits while constructing the approximate solution.

Theory and Semi-Analytical Study of Micropolar Fluid Dynamics through a Porous Channel

In this work, We are looking at the characteristics of micropolar flow in a porous channel that’s being driven by suction or injection. The working of the fluid is described in the flow model. We can reduce the governing nonlinear partial differential equations (PDEs) to a model of coupled systems of nonlinear ordinary differential equations using similarity variables (ODEs). In order to obtain the results of a coupled system of nonlinear ODEs, we discuss a method which is known as the differential transform method (DTM). The concern transform is an excellent mathematical tool to obtain the analytical series solution to the nonlinear ODEs. To observe beast agreement between analytical method and numerical method, we compare our result with the Rung-Kutta method of order four (RK4). We also provide simulation plots to the obtained result by using Mathematica. On these plots, we discuss the effect of different parameters which arise during the calculation of the flow model equations.