Form Analytical Solution Research Articles

Time-varying relative risk aversion: Theoretical mechanism and empirical evidence

This paper explores the issue of understanding time-varying relative risk aversion with household-level data on two classical portfolio choice problems. First, we derive an analytic form solution to a parsimonious portfolio choice model with the preference given by Greenwood, Hercowitz and Huffman (1988, GHH), and then, the solution identifies four partial equilibrium effects in our model with the GHH preference on risky shares through two channels and two net effects whose signs hinge on the value of a key structural parameter. Based on household-level data, our empirical results from both mean and quantile regression models show clearly that wealth negatively affects risky shares and the estimated effects are statistically significant and robust, which is in line with the theory. Finally, we show that the GHH preference alone is not sufficient in explaining how risky shares respond to labor income in the household-level data.

Sound diffraction by knife-edges of finite length: Integral solution, dimensionless parameters, and explicit formulas.

Sound diffraction by knife-edges of finite length is considered in the frequency domain. An approximate analytical solution in integral form is derived from a previously published time domain solution. Unlike the well-established finite length diffraction solution by Svensson et al. [Acta Acust. Acust. 95(3) 568-572 (2009)], the presented solution contains no singularities and both solutions agree, except very close to the diffracting edge. It is shown that finite length diffraction can be studied based on two dimensionless parameters: one expressing the receiver's proximity to the shadow boundary and one associated with the edge length. Depending on the dimensionless parameters, a given edge can be considered a short edge, an infinitely long edge or an edge of medium length, each case with different characteristics. Furthermore, a nomograph and the corresponding database are presented. They provide the normalized diffracted field for any source/receiver location, any source frequency, and any edge length. Also, easy to compute explicit mathematical expressions are presented to approximate the analytical integral solution. These expressions, along with the database method, accelerate diffraction calculations by order of magnitude compared to the presented integral solution or the Svensson solution. Finally, predictions from all proposed methods agree reasonably well with experimental data.

Hybrid Model for Optimisation of Waste Dump Design and Site Selection in Open Pit Mining

Waste management is an unavoidable technological operation in the process of raw material extraction. The main characteristic of this technological operation is the handling of large quantities of waste material, which can amount to several hundred million cubic metres. At the same time, this operation must comply with all administrative and environmental standards. Therefore, optimising waste rock management (particularly haulage and dumping) has the potential to significantly improve the overall value of the project. This paper presents a hybrid model for the optimisation of waste dump design and site selection. The model is based on different mathematical methods (Monte Carlo simulation, genetic algorithm, analytic hierarchy process and heuristic methods) adapted to different aspects of the problem. The main objective of the model is to provide a solution (in analytical and graphical form) for the draft waste dump design, on the basis of which the final waste dump design can be defined. The functioning of the model is verified using an example of an existing open pit. In the case study, 2250 members of the initial population (different waste dump variants) were generated, and a total of 110 optimised solutions were obtained using 15 optimisations. The solution with the best value of the objective function is adopted, and the final waste dump design is created.

Analysis of functionally graded rotating disks via analytical approximation methods

Analytical solutions for functionally graded (FG) rotating disks are limited due to the functional representation of the cross-sectional geometry, the density, and the modulus of elasticity which lead to a governing differental equation with variable coefficients. Analytical solutions for such disks are limited according to the functional form chosen for the variation of the property. Hence, analytical approximation approaches gain importance for such problems. In this study, two analytical approximation approaches are proposed for the analysis FG rotating disks. The methods used are the Improved Adomian decomposition method (IADM) and the optimal homotopy asymptotic method (OHAM). The resulting solution is an analytical solution in power series form. It is assumed that the disks considered are subjected to constant angular velocity. Material properties such as modulus of elasticity, density and yield strength are assumed to vary along the radial direction according to the power law function. Stresses are obtained for different values of the gradient parameters to investigate the effects of the parameters. Von Mises yield criterion is used to determine elastic limit angular velocities. The results from this study are in very good agreement with previous results available in the literature. In addition, it is observed that OHAM converges to results much faster than IADM.

A modern analytic method to solve singular and non-singular linear and non-linear differential equations

This article circumvents the Laplace transform to provide an analytical solution in a power series form for singular, non-singular, linear, and non-linear ordinary differential equations. It introduces a new analytical approach, the Laplace residual power series, which provides a powerful tool for obtaining accurate analytical and numerical solutions to these equations. It demonstrates the new approach’s effectiveness, accuracy, and applicability in several ordinary differential equations problem. The proposed technique shows the possibility of finding exact solutions when a pattern to the series solution obtained exists; otherwise, only rough estimates can be given. To ensure the accuracy of the generated results, we use three types of errors: actual, relative, and residual error. We compare our results with exact solutions to the problems discussed. We conclude that the current method is simple, easy, and effective in solving non-linear differential equations, considering that the obtained approximate series solutions are in closed form for the actual results. Finally, we would like to point out that both symbolic and numerical quantities are calculated using Mathematica software.

Optimal designs for prediction in random coefficient regression with one observation per individual

The subject of this work is random coefficient regression models with only one observation per observational unit (individual). An analytical solution in form of optimality conditions is proposed for optimal designs for the prediction of individual random effect for a group of selected individuals. The behavior of optimal designs is illustrated by the example of linear regression models.

AN ANALYTICAL STUDY OF SPACE-TIME FRACTIONAL ORDER GAS DYNAMIC EQUATIONS

In this article, the Sumudu transform with iterative method is implemented to obtain approximate analytical solutions in series form to non-linear homogeneous and non-homogeneous space-time fractional gas dynamic equations. The fractional derivatives presented here are in the Caputo sense. Furthermore, the findings of this study are graphically represented and the solution graphs demonstrate a strong connection between the approximate and exact solutions.

A reliable approach of Ramadan group integral and projected differential transform methods to system of nonlinear partial differential equations

In this paper, a hybrid method is presented via combination of the Ramadan Group Integral method (RGITM) with method of Projected Differential Transform(PDTM) for the purpose of solving nonlinear partial differential equations systems. The method?s goal is to produce analytical solutions in series form. In comparison to existing methods, the suggested method makes handling such partial differential equations simple. The outcome demonstrated the method?s effectiveness, accuracy, and validity. The technique can be easily applied to a wide variety of nonlinear issues, and it has the potential to both reduce the amount of computation required and deal with the flaw brought about by the nonlinear components that cannot be resolved by employing recognized integral transforms. Examples will be looked at to help illustrate the proposed analysis.

Application of Non-singular Kernel in a Tumor Model with Strong Allee Effect

We obtain the analytical solutions in implicit form of a tumor cell population differential equation with strong Allee effect. We consider the ordinary case and then a fractional version. Some particular cases are plotted.

On spectral polar fractional Laplacian

In this paper, we introduce the fractional Sturm–Liouville operators and present the polar fractional Laplacian as a particular case of these operators. We also consider the distributed order space–time fractional diffusion equation involving the polar fractional Laplacian and propose three approaches for studying this problem on the circle and ring domains. First, we apply the integral transforms as the analytical methods to get the solution in terms of the Laplace-type integrals using the Titchmarsh theorem. Second, we use the finite difference method for discretization of the space variable and employ the matrix transfer technique to obtain a system of the distributed order fractional equations. For this system, we modify the Putzer’s algorithm and get the solution in terms of the eigenvalues of discretization matrix. Third, we employ the backward Euler numerical method for discretization of the time variable and find the corresponding error bound. Moreover, we compare and verify the approximate solutions with the exact solutions in analytical form.

Effect of Catheter and Stenosis on Solute Diffusion in Non-Newtonian Blood Flow through a Catheterized Stenosed Artery

The presence of stenosis at the wall of the artery lead to further cardiovascular diseases such as heart attack, stroke and many more. Treatment of a stenosed artery includes the insertion of a catheter through the artery which affects the blood flow and solute dispersion. This present study focuses on the effect of catheter radius and stenosis height on the blood flow and solute dispersion behavior. The problem is modelled using the Herschel-Bulkley fluid to represent the blood rheology, with catheter and stenosis as the boundary conditions. Analytical solutions in integral form are obtained by solving the momentum equation and Herschel-Bulkley constitutive equation. The integrals are numerically evaluated using the Simpson’s 3/8 rule and Regula-Falsi method to obtain the blood velocity. The obtained velocity is employed into the unsteady convective-diffusion equation and solved using the generalized dispersion model (GDM) to analyse the behaviour of solute diffusion. The influence of catheter radius and stenosis height on the diffusion coefficient and mean concentration of solute are observed. Results show that the diffusion coefficient decreases as the catheter radius and stenosis height increases. A decrease in diffusion coefficient simultaneously increases the solute mean concentration.

Development and theoretical analysis of slippery walls flow of third-grade fluid through the convergent symmetric channel

The present investigation is a study of third-grade fluid muddled with tiny-sized round-shaped hafnium particles drifting through a convergent geometry under the influence of slip boundary conditions. The impact of electro-osmotic is also examined in this investigation. The Debye Hückel linearization is used to investigate the current problem. Fluid and particulate phases are illustrated in convergent geometry having momentous utilizations in industry and different strolls of science. The problem is modeled in terms of nonlinear complex differential equations by using the fluid and particle phase models. The normalized quantities are used to convert the dimensional equations into dimensionless form and the perturbation method is selected to obtain the solution in analytical form. The effects of pertinent parameters on each type of flow are also presented graphically. Promotion in the velocity profile is observed for the volumetric flow parameter. The slip parameter boost motion of the fluid particle inside the channel. The current study can be utilized in the chemical industry, reservoir engineering, and microfabrication technique in which mass exchanges and electro-osmotic energy play a vital role.

Subharmonic Resonance of Duffing Oscillator With Dry Friction Under Foundation Excitation

Abstract The 1/3 subharmonic resonance response of Duffing oscillator with Coulomb dry friction under foundation excitation is investigated, and the approximate analytical solution of the subharmonic resonance of the system is obtained by using the incremental averaging method. Based on the approximate analytical solution of the primary resonance obtained by the averaging method, the approximate analytical solution of subharmonic resonance is solved by the averaging method according to the incremental equation, and the amplitude–frequency response equation of subharmonic resonance is obtained. It is found that the Coulomb friction affects the amplitude–frequency response of both the primary resonance and subharmonic resonance of the nonlinear dry friction system in the form of equivalent damping. The comparison between the approximate analytical solution and the numerical solution shows that the approximate analytical solutions of the primary resonance and subharmonic resonance are both in very good agreement with the numerical solution. The existence condition of the 1/3 subharmonic resonance for the nonlinear dry friction system is presented, and the stability of the steady-state solution of subharmonic resonance is also judged. Based on the approximate analytical solution, the effects of the nonlinear stiffness and the Coulomb friction on the amplitude–frequency response of resonance and critical frequency of 1/3 subharmonic resonance of the nonlinear dry friction system are analyzed in detail. The analysis results show that the incremental averaging method can effectively obtain the approximate analytical solution in unified form for the subharmonic resonance of nonlinear system with Coulomb friction.

Forecasting the behaviour of fractional Black-Scholes option pricing equation by laplace perturbation iteration algorithm

Financial derivatives plays a major role in all financial deals these days. Black–Scholes option pricing model gives a risk free analysis for investing in options. In the current work, a method called the Laplace Perturbation Iteration Algorithm is being applied on Fractional Black–Scholes Equation to obtain its fractional analytical solutions in series form to analyze its results for the European and American option pricing problem, quickly and accurately. Laplace Perturbation Iteration Algorithm incorporates Laplace transform and Perturbation Iteration Algorithm and forms an iterative scheme that derives the analytical solutions without any inconvenience. The Fractional Black–Scholes Equation analytical solution is obtained as a convergent power series of conveniently computed components. Four examples have been solved and their results have been discussed in this paper, in order to check the efficacy of Laplace Perturbation Iteration Algorithm.

On the influence of the pipe outer coating nonuniformity on the contact stresses arising under rigid bushes

The work is devoted to the formulation and construction of a solution in analytical form for the mechanical problem of contact interaction between a finite system of rigid cylindrical bushes and a pipe with an external coating, which has longitudinal nonuniformity. In industry and mechanical engineering, the presence of such additional coatings is required, for example, for protection the pipe main layer from external aggressive media, for its electrical or thermal insulation, etc. The nonuniformity of such a layer may be due to various additional requirements, as well as the obtaining process of this layer. When analyzing the stress-strain state of the pipe interacting with the bushes, it is necessary to take into account all the features. The paper shows that the mathematical model of the above-described interaction problem is a system of mixed integral equations containing integral operators of different types. Moreover, the resulting system includes functions describing the variable properties of the outer layer in the area of interaction with the bushings. The special approach described by the main theses in this paper makes it possible to obtain a solution in a form in which all complex functions are allocated as separate multipliers. This makes it possible to perform calculations even in cases where the coating is highly heterogeneous or consists of different materials.

Analytical Approach to Fractional Fisher Equations by Laplace-Adomian Decomposition Method

This article implements the Laplace-Adomian decomposition method to obtain approximate analytical solutions in series form for non-linear time-fractional Fisher's equations with initial conditions. The fractional derivatives are given in the sense of Caputo. In addition, the results of this investigation are represented graphically, and they are simple yet highly accurate and compare favourably with the solutions reported in the earliest literature.

Analytical solution for arbitrary large deflection of geometrically exact beams using the homotopy analysis method

Beam-like compliant elements have found wide-ranging application in many fields of engineering and science often where 3D large deflections can be of concern such as soft robotics, DNA mechanics and helicopter/wind turbine rotor blades. The homotopy analysis method (HAM) is used for the first time to obtain a novel analytical solution in converged series form for the arbitrary large deflection of geometrically exact beams subject to both conservative and follower loading scenarios. The homotopy analysis method, which offers desirable characteristics such as being free from small or large parameters, coupled with auxiliary parameters controlling convergence, is applied directly to the intrinsic governing equations of a geometrically exact beam theory. The system of first-order differential governing equations of geometrically exact beams with intrinsic formulation is free from rotation and displacement variables, and offers a low degree of nonlinearity (quadratic at most) and compact mathematical form, making it suitable for analytical solutions. Due to the relatively poor convergence of the original HAM algorithm, the iterative HAM technique is employed which is known to accelerate convergence and to improve the computational efficiency of the homotopy analysis method. The obtained homotopy series offers a number of novel features in the context of the analytical solutions for the large deflection of beams, including (a) the direct calculation of internal forces and moments which is significant for engineering design purposes, (b) being able to capture 3D deflections, (c) considering transverse shear effects which can be important for thicker beams or when the Young’s modulus to shear modulus ratio is significant (such as composite materials) and (d) considering conservative and follower tip and distributed loads, in a unified framework. In order to investigate the efficacy, applicability and accuracy of the proposed method, a number of numerical examples are considered in which a cantilever beam subject to tip or distributed loads undergoes large deflection. Large deflection results for both conservative and follower loads are compared against those of less comprehensive analytical solutions as well as against numerical methods including finite element and Chebyshev collocation methods where good agreement is observed. These results demonstrate the applicability and effectiveness of HAM for the large deflection analysis of geometrically exact beams.

Analytical solution of systems of Volterra integro-differential equations using modified differential transform method

In this paper, a hybrid technique based on the combining of differential transform approach, the Laplace transformation, and Pade approximant is successfully implemented for solving Volterra Integro differential (VID) equations subject to appropriate conditions. The proposed method known as modified differential transform method (MDTM) enlarges the domain of convergence with less computational time. One of the interesting features of this algorithm is the ability to produce an analytic approximate solution in convergent power series form with few numbers of computable terms. Preliminary results on some test problems show that the new algorithm is efficient and promising for Volterra integro-differential equations and can be applied as alternatives for the integral and integro-differential (IID) equation that have no analytical solution.

New insights into the drainage of inundated ice-wedge polygons using fundamental hydrologic principles

Abstract. The pathways and timing of drainage from the inundated centers of ice-wedge polygons in a warming climate have important implications for carbon flushing, advective heat transport, and transitions from methane to carbon dioxide dominated emissions. Here, we expand on previous research using a recently developed analytical model of drainage from a low-centered polygon. Specifically, we perform (1) a calibration to field data identifying necessary model refinements and (2) a rigorous model sensitivity analysis that expands on previously published indications of polygon drainage characteristics. This research provides intuition on inundated polygon drainage by presenting the first in-depth analysis of drainage within a polygon based on hydrogeological first principles. We verify a recently developed analytical solution of polygon drainage through a calibration to a season of field measurements. Due to the parsimony of the model, providing the potential that it could fail, we identify the minimum necessary refinements that allow the model to match water levels measured in a low-centered polygon. We find that (1) the measured precipitation must be increased by a factor of around 2.2, and (2) the vertical soil hydraulic conductivity must decrease with increasing thaw depth. Model refinement (1) accounts for runoff from rims into the ice-wedge polygon pond during precipitation events and possible rain gauge undercatch, while refinement (2) accounts for the decreasing permeability of deeper soil layers. The calibration to field measurements supports the validity of the model, indicating that it is able to represent ice-wedge polygon drainage dynamics. We then use the analytical solution in non-dimensional form to provide a baseline for the effects of polygon aspect ratios (radius to thaw depth) and coefficient of hydraulic conductivity anisotropy (horizontal to vertical hydraulic conductivity) on drainage pathways and temporal depletion of ponded water from inundated ice-wedge polygon centers. By varying the polygon aspect ratio, we evaluate the relative effect of polygon size (width), inter-annual increases in active-layer thickness, and seasonal increases in thaw depth on drainage. The results of our sensitivity analysis rigorously confirm a previous analysis indicating that most drainage through the active layer occurs along an annular region of the polygon center near the rims. This has important implications for transport of nutrients (such as dissolved organic carbon) and advection of heat towards ice-wedge tops. We also provide a comprehensive investigation of the effect of polygon aspect ratio and anisotropy on drainage timing and patterns, expanding on previously published research. Our results indicate that polygons with large aspect ratios and high anisotropy will have the most distributed drainage, while polygons with large aspect ratios and low anisotropy will have their drainage most focused near their periphery and will drain most slowly. Polygons with small aspect ratios and high anisotropy will drain most quickly. These results, based on parametric investigation of idealized scenarios, provide a baseline for further research considering the geometric and hydraulic complexities of ice-wedge polygons.

Application of hybrid asymptotic methods and modern software for mathematical models developing for nonlinear dynamics of structures with variables in time parameters

The relevance. The aerospace domain requires studies of mathematical models of nonlinear dynamic structures with time-varying parameters.
 The aim of the work. To obtain an approximate analytical solution of nonlinear forced oscillations of the designed models with time-dependent parameters.
 The research methods. A hybrid approach based on perturbation methods, phase integrals, Galorkin orthogonalization criterion is used to obtain solutions.
 Results. Nonlocal investigation of nonlinear systems behavior is done using results of analytical and numerical methods and developed software. Despite the existence of sufficiently powerful numerical software systems, qualitative analysis of nonlinear systems with variable parameters requires improved mathematical models based on effective analytical, including approximate, solutions, which using numerical methods allow to provide a reliable analysis of the studied structures at the stage designing. An approximate solution in analytical form is obtained with constant coefficients that depend on the initial conditions.
 Conclusions. The approximate analytical results and direct numerical solutions of the basic equation were compared which showed a sufficient correlation of the obtained analytical solution. The proposed algorithm and program for visualization of a nonlinear dynamic process could be implemented in nonlinear dynamics problems of systems with time-dependent parameters.