Infinite Series Solution Research Articles

Infinite series solution of the time-dependent radiative transfer equation in anisotropically scattering media

We solve the radiative transfer equation (RTE) in anisotropically scattering media as an infinite series. Each series term represents a distinct number of scattering events, with analytical solutions derived for zero and single scattering. Higher-order corrections are addressed through numerical calculations or approximations.The RTE solution corresponds to Monte Carlo sampling of photon trajectories with fixed start and end points. Validated against traditional Monte Carlo simulations, featuring random end points, our solution demonstrates enhanced efficiency for both anisotropic and isotropic scattering functions, significantly reducing computational time and resources. The advantage of our method over Monte Carlo simulations varies with the position of interest and the asymmetry of light scattering, but it is typically orders of magnitude faster while achieving the same level of accuracy. The exploitation of hidden symmetries further accelerates our numerical calculations, enhancing the method’s overall efficiency.In addition, we extend our analysis to the first and second moments of the photon’s flux, elucidating the transition between transport and diffusive regimes.

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

Analytical modeling of phase change in a composite wall comprising two distinct phase change materials in series

Most of the past literature on solid-liquid phase change heat transfer modeling focuses on a single phase change material (PCM), whereas, a sandwich of two or more PCMs arranged in series may be of interest in several applications. The modeling of heat transfer and phase change in such a system is complicated by the presence of multiple phase change fronts propagating at the same time. This work presents a theoretical analysis of the problem of melting of a two-PCM stack being heated from one end. Depending on whether the lower-melting PCM is located next to or away from the heat source, two distinct cases are considered. The propagation of the phase change fronts in each case is divided into several stages, each of which is characterized by a distinct phase change and/or sensible heating processes. The transient temperature field in each stage is determined in the form of infinite series solutions, and the propagation of phase change front over time is then determined using energy conservation at the phase change interface. In this manner, explicit expressions are derived for the time taken for each layer to completely melt. Results are shown to be in good agreement with finite-element simulations, and with the exact Stefan solution under special conditions. Key non-dimensional parameters that influence the nature of the two-PCM phase change process are identified, and their impact on phase change performance is analyzed. Quantitative predictions are presented for the conditions under which both PCMs finish melting at the same time, which is a favorable outcome for efficient energy storage.

Modeling the frequency response of an acoustic cavity using the method of images

We demonstrate the ability of a simple algorithm based on the venerable method of images (MOIs), to accurately model the detailed frequency response of a multidimensional, rectangular, lossy resonant cavity. The convergence properties of the model's infinite series solution are shown to be determined by the cavity's quality factor Q. A 1D example demonstrates that the MOI series converges to the exact solution. Next, a comparison to precisely measure 2D cavity data confirms that a straightforward extension of the 1D algorithm to multiple dimensions provides accurate results. The algorithm is short, easily understandable by undergraduate students and relatively undemanding to code. An example using ®mathematica is provided.

Construction of Infinite Series Exact Solitary Wave Solution of the KPI Equation via an Auxiliary Equation Method

The KPI equation is one of most well-known nonlinear evolution equations, which was first used to described two-dimensional shallow water wavs. Recently, it has found important applications in fluid mechanics, plasma ion acoustic waves, nonlinear optics, and other fields. In the process of studying these topics, it is very important to obtain the exact solutions of the KPI equation. In this paper, a general Riccati equation is treated as an auxiliary equation, which is solved to obtain many new types of solutions through several different function transformations. We solve the KPI equation using this general Riccati equation, and construct ten sets of the infinite series exact solitary wave solution of the KPI equation. The results show that this method is simple and effective for the construction of infinite series solutions of nonlinear evolution models.

Impact of Inclined Magnetic Force on Bio-Fluid in Permeable Bifurcated Arteries: Analytical Approach

The present article aims at presenting analytical solutions of the effects of chemical reaction and inclined magnetic force on blood flow through bifurcated arteries placed in a porous medium in association with heat source. The equations of the blood flow model are solved analytically by means of infinite series solution of convergent scheme with appropriate initial and boundary conditions. The important characteristics of the electromagnetohydrodynamic flow of bio-fluid through bifurcated arteries are distinctly highlighted by virtue of the dual solutions that are obtained for Axial velocity, Normal velocity, fluid Temperature, molar species, Skin-friction, Nusselt Number and Sherwood Number. The behaviour of the biofluid variables with individual parameters like Prandtl Number (Pr), Magnetic drag force (Md), Porosity (Kp), Heat source (Hs), Schmidt Number (Sc), Thermal Radiation (R), Chemical reaction rate (Cr), decay (ξ) are discussed in detail through graphs using MATLAB Software. Validation of this work is presented and found suitable. It is disclosed that the flow of bio-fluid is noticeably influenced by the adequate strength of externally applied inclined magnetic force and porosity. This study is essentially important in simple flow, peristaltic flow, pulsatile flow and drug delivery.

Construction of New Infinite-Series Exact Solitary Wave Solutions and Its Application to the Korteweg–De Vries Equation

The Korteweg–de Vries (KDV) equation is one of the most well-known models in nonlinear physics, such as fluid physics, plasma, and ocean engineering. It is very important to obtain the exact solutions of this model in the process of studying these topics. In the present paper, using distinct function iteration relations in two ways, namely, squaring infinitely and extracting the square root infinitely, which have not been reported in other documents, we construct abundant types of new infinite-series exact solitary wave solutions using the auxiliary equation method. Most of these solutions have not been reported in previous papers. The numerical analysis of some solutions shows complex solitary wave phenomena. Some solutions can have stable solitary wave structures, while others may have singularities in certain space–time positions. The results show that the analysis model we use is very simple and effective for the construction of new infinite-series solutions and new solitary wave structures of nonlinear models.

Ramanujan: The New Sum of All Natural Numbers

Abstract: As we know that Sir Ramanujan gave the solution of sum of all natural numbers up to infinity and said that the sum of all natural numbers till infinity is -1/12. I studied on this topic and found that if we try to solve the infinite series in a slightly different way, then we get the answer of its sum different from -1/12, so this is what I have written in this paper that such Ramanujan Sir, what was the mistake in solving the infinite series, which by solving it in a slightly different way from the same concept, we get different answers. Keywords: Ramanujan the new sum of all natural numbers, Infinite series solution, Gaurav singh patel, Infinite series sum, Natural number sum.

Two fast and accurate routines for solving the elliptic Kepler equation for all values of the eccentricity and mean anomaly

Context. The repetitive solution of Kepler’s equation (KE) is the slowest step for several highly demanding computational tasks in astrophysics. Moreover, a recent work demonstrated that the current solvers face an accuracy limit that becomes particularly stringent for high eccentricity orbits. Aims. Here we describe two routines, ENRKE and ENP5KE, for solving KE with both high speed and optimal accuracy, circumventing the abovementioned limit by avoiding the use of derivatives for the critical values of the eccentricity e and mean anomaly M, namely e > 0.99 and M close to the periapsis within 0.0045 rad. Methods. The ENRKE routine enhances the Newton-Raphson algorithm with a conditional switch to the bisection algorithm in the critical region, an efficient stopping condition, a rational first guess, and one fourth-order iteration. The ENP5KE routine uses a class of infinite series solutions of KE to build an optimized piecewise quintic polynomial, also enhanced with a conditional switch for close bracketing and bisection in the critical region. High-performance Cython routines are provided that implement these methods, with the option of utilizing parallel execution. Results. These routines outperform other solvers for KE both in accuracy and speed. They solve KE for every e ∈ [0, 1 − ϵ], where ϵ is the machine epsilon, and for every M, at the best accuracy that can be obtained in a given M interval. In particular, since the ENP5KE routine does not involve any transcendental function evaluation in its generation phase, besides a minimum amount in the critical region, it outperforms any other KE solver, including the ENRKE, when the solution E(M) is required for a large number N of values of M. Conclusions. The ENRKE routine can be recommended as a general purpose solver for KE, and the ENP5KE can be the best choice in the large N regime.

Theoretical investigation on submicron particle penetration through circular tubes inside a porous medium

The analytical infinite series solution of submicron particle transport in a circular tube bounded by a porous wall, such as a pinhole, is determined under the slip velocity boundary condition, and the solution is verified by using the experimental data in the previous studies for the specific cases. The results show that particle penetration rate increases with the increase of the porous parameter, the axial pressure drop, and the pinhole radius, whereas it decreases with increasing the pinhole length. The penetration rate of nano-particles are more sensitive to the variation of these parameters. However, the differences between the penetrations of particles ranging from 0.3 μm to 1 μm are not evident because the diffusion becomes weak gradually in this size range. In addition, a further comparison is performed between the analytical solution and the existing studies, and approximate expressions are presented for accurate calculation of particle penetration rate through pinholes appearing in porous materials including filter devices and masks.

A new method to solve electrolyte diffusion equations for single particle model of lithium-ion batteries

It is one of basic tasks to solve the electrochemical model of lithium-ion batteries for obtaining the lithium-ion concentration in the electrolyte. In order to balance the computational efficiency and electrolyte dynamic property, it is assumed that reactions occur only at interfaces between the collector and the electrolyte. Based on the analytical solution to the liquid diffusion equations, which is in the form of infinite series, a new method is proposed to solve it. Under galvanostatic profiles, the analytic solution is an infinite time series transformed into a converged sum function by using the monotone convergence theorem. Under the dynamic profiles, the infinite series solution is simplified into an infinite discrete convolution of both the input and the sum function. The sum function is truncated by its characteristic of monotonic decay approaching to zero over time, thus obtaining a finite discrete convolution algorithm. Reference to the results from a professional finite element analysis software, the proposed algorithm can produce high accuracy with less computation time under both galvanostatic profiles and dynamic profiles. Also, there is only one parameter to be configured. Therefore, our algorithm will reduce the computation burden of the electrochemical model applied to a real-time battery management system.

Efficient Solution of Fractional-Order SIR Epidemic Model of Childhood Diseases With Optimal Homotopy Asymptotic Method

In providing an accurate approximate analytical solution to the non-linear system of fractional-order susceptible-infected-recovered epidemic model (FOSIREM) of childhood disease has been a challenge, because no norm to guarantees the convergence of the infinite series solution. We compute an accurate approximate analytical solution using the optimal homotopy asymptotic method (OHAM). The fractional differential equations operator (FDEO) is given as conformable derivative operator (CDO). We show the basic idea of the proposed method, the CDO sense, equilibrium points, local asymptotic stability, reproduction number, and the convergence analysis of the proposed method. Numerical results and comparisons with other approximate analytical methods are given to validate the efficiency of the method. The proposed method speedily converges to the exact solution as the fractional-order derivative approaches 1, proved as an excellent tool for solving, and predicting the model.

A new solution to the spherical particle surface concentration of lithium-ion battery electrodes

The lithium-ion concentration of a solid phase is essential to solve the electrochemical model of a lithium-ion battery. Based on the analytic solution of a convolution infinite series, a new algorithm is proposed to efficiently and accurately solve the partial differential equation for the lithium-ion diffusion behavior of electrode particles. Under galvanostatic profiles, the analytic solution is an infinite time series transformed into a converged sum function by using the monotone convergence theorem. Under dynamic profiles, the infinite series solution is simplified to a finite discrete convolution of both the input and the sum function. Meanwhile, the discrete step is determined by the amplitude-frequency characteristic curve, and the sum function is truncated by its characteristic monotonic decay approaching zero over time. Compared with using a professional finite element analysis software, it takes less time to use this discrete convolution algorithm, and it yields less error. And, it has the least solution time with medium accuracy in comparison with two commonly used approximations. Not only that, the proposed algorithm has only two parameters to be determined with little computation, thus promoting the electrochemical models built in battery management systems.

Semi-analytical solutions for buckling and free vibration of composite anisogrid lattice cylindrical panels

In this study, highly accurate approximate closed-form solution is presented for buckling and free vibration of clamped composite anisogrid lattice cylindrical panels. Using theory of orthotropic cylindrical shallow shells based on the continuous model of lattice structures, the system of governing partial differential equations (PDEs) of motion is derived. The resulted PDEs are converted to a couple of systems of ordinary differential equations (ODEs) by employing of the Extended Kantorovich Method (EKM). Semi-analytical solutions are presented for both buckling and free vibration of anisogrid lattice panels via iterative application of infinite power series solution for the sets of ODEs. Efficiency of the presented method in terms of accuracy, stability, fast convergence and low computational cost for prediction of the critical buckling loads and natural frequencies are examined. Accuracy of the predictions for various cases are studied by comparing with available numerical and analytical results. Moreover, it is shown that the method can be used to obtain accurate solutions for buckling and free vibration of composite lattice rectangular plates by assuming large values for the radius together with small values for the angle of the panel. This implies stability of the method for large or small values while providing accurate predictions.

Analysis of Platelet Shape Al2O3 and TiO2 on Heat Generative Hydromagnetic Nanofluids for the Base Fluid C2H6O2 in a Vertical Channel of Porous Medium

An analytical investigation is performed on the unsteady hydromagnetic flow of nanoparticles Al2O3 and TiO2 in the EG base fluid through a saturated porous medium bounded by two vertical surfaces with heat generation and no-slip boundary conditions. The physics of initial and boundary conditions is designated with the flow model's non-linear partial differential equations. The analytical expressions of nanofluid velocity and temperature with the channel are derived, and Matlab Codes are used to plot the significant results for physical variables. From the physical point of view for nanofluid velocity and temperature results, the base fluid C2H6O2 has a higher viscosity and thermal conductivity than that of water. Physically, the platelet shape Al2O3 nanofluid has the highest velocity than TiO2 nanofluid. It is found that the velocity of nanofluid enhanced the porosity and nanoparticles volume fraction for Al2O3 - EG and TiO2 - EG base nanofluids. However, this trend is reversed for the effects of heat generation. Obtained results indicate that an increase in nanoparticles volume fraction raises the skin friction near the surface, but profiles gradually become linear, due to less frictional effects of nanoparticles. Moreover, due to higher values of nanoparticles volume fraction, the thermal conductivity is raised, and thus the thickness of the thermal boundary layer is declined. The results show that the method provides excellent approximations to the analytical solution of nonlinear system with high accuracy. Metal oxide nanoparticles have wide applications in various fields due to their small sizes, such as the pharmaceutical industry and biomedical engineering. HIGHLIGHTS Impact of platelet shape Al2O3 and TiO2 for base fluid C2H6O2 is studied In Couette and Poiseuille flow, nanoparticles play a vital role to enhance the heat transfer The infinite series solution has been used for solving the non-linear PDE’s The uses of Al2O3 and TiO2 in significant heat transfer applications is overviewed The physiochemical and structural features of metal oxide nanoparticles have diverse biomedical applications GRAPHICAL ABSTRACT

Exact solution to a multidimensional wave equation with delay

This paper deals with a mixed problem for the wave equation with discrete delay τ>0,utt(t,x)=a12Δxu(t,x)+a22Δxu(t−τ,x)+b1u(t,x)+b2u(t−τ,x),t>τ,0≤x≤l,with Dirichlet boundary conditions. The exact infinite series solution is constructed by the method of separation of variables, where the time-dependent functions of the decomposition satisfy second-order delay differential equations. Our approach is based on and extends the work by Rodríguez, Roales and Martín (Applied Mathematics and Computation, 2012).

An improved single-particle model with electrolyte dynamics for high current applications of lithium-ion cells

In this work, an isothermal, reduced-order polynomial-based model is proposed for lithium-ion cells. The correction incorporated into the single-particle model considers the effect of electrolyte dynamics leading to improved model predictions at high current applications without adding any significant computational complexity. The model includes the spatial distribution of the electrode and electrolyte potentials and the electrolyte lithium concentration on the resulting cell voltage. A novel approach to account for the spatially varying overpotential and open-circuit potential is proposed. A linear system of differential-algebraic equations is obtained in the state variables and the cell voltage is computed using a non-linear expression in these states. The resulting linear state-space system makes the model easily implementable for state-estimation algorithms in battery management system applications. Additionally, a semi-analytical model is incorporated for the diffusion of lithium in electrodes and a criterion for truncating the infinite-series solution is proposed for achieving a balance between accuracy and complexity for cells subjected to fluctuating currents. The proposed model results in a decrease in error for cell voltage prediction by a factor of five when compared with existing enhanced single-particle models.

Thermoelastic response of a nonhomogeneous elliptic plate in the framework of fractional order theory

This article aims to examine the thermoelastic problem for a medium with nonhomogeneous material properties within a finite-length elliptical plate. An analytical method is established under the assumption that the thermomechanical material inhomogeneity obeys the product form of power and exponential functions through the plate's thickness. At the same time, calorific capacity and Poisson's ratio are assumed to be constant. The fractional time derivative heat conduction differential equation with an internal heat source is solved using integral transformation technique in terms of generalized Laguerre polynomial, and Mathieu function in the Laplace domain. The Gaver–Stehfest approach is used to invert Laplace domain outcomes. The convergence of infinite series solutions has been discussed. In the specific case, the elliptic region that degenerates into the problem of the circular area by adding limiting conditions is also considered. As a numerical example, the temperature, displacement and thermal stress distributions are calculated, taking into account influence of inhomogeneity, and the numerical results are graphically demonstrated.

Bivariate Infinite Series Solution of Kepler’s Equations

A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This result is based on an iterative procedure for the analytical computation of all the higher-order partial derivatives of the eccentric anomaly with respect to the eccentricity e and mean anomaly M in a given base point (ec,Mc) of the (e,M) plane. Explicit examples of such bivariate infinite series are provided, corresponding to different choices of (ec,Mc), and their convergence is studied numerically. In particular, the polynomials that are obtained by truncating the infinite series up to the fifth degree reach high levels of accuracy in significantly large regions of the parameter space (e,M). Besides their theoretical interest, these series can be used for designing 2-D spline numerical algorithms for efficiently solving Kepler’s equations for all values of the eccentricity and mean anomaly.

Homotopy analysis method approximate solution for fuzzy pantograph equation

This paper investigates the powerful method namely the homotopy analysis method (HAM), to solve the fuzzy pantograph equation (FPE) in approximate analytic form. HAM yields a convergent infinite series solution to the solution of FPE without the need to reduce the FPE to the first order system or compare it with the exact solution, and this is one of the advantages of this method. For a better approximate solution, the HAM uses a convergence control parameter from the convergence region of the infinite series solution. HAM solution of FPE is obtained by reformulate crisp standard approximation via the properties of the fuzzy set theory.