Separation Of Variables Technique Research Articles

Solution of fractional Laplace type equation in conformable sense using fractional fourier series with separation of variables technique

Solution of fractional Laplace type equation in conformable sense using fractional fourier series with separation of variables technique

Service life of RC seawall under chloride invasion: A theoretical model incorporating convection-diffusion effect

This paper presents a theoretical approach, which comprises the chloride transportation model and cover cracking model, to predict the service life of RC seawall under chloride invasion. The chloride transportation model combined the Fick's second law and Darcy's law to model the convection-diffusion of chloride ions in the RC seawall. The governing equation for chloride migration in the RC seawall is solved with the separation of variables technique. The proposed model is validated by comparing with a corresponding finite difference solution and the experimental results. The mechanical model that represents the cracking of the concrete cover induced by the expansion pressure of the corrosion products is finally developed to predict the service life of the RC seawall under chloride invasion. Parametric studies are performed to illustrate the effects of convection-diffusion and other key variables on the chloride ions transportation and service life of the RC seawall. The results reveal that the convection effect substantially accelerates the chloride penetration process in the chloride gradient descent direction while inhibits the penetration in the opposite direction. The proposed analytical solution in conjunction with the mechanical model provides a theoretical approach for predicting the service life of RC seawall, which can be used to aid durability design of coastal structures.

On a Solution of a Third Kind Mixed Integro-Differential Equation with Singular Kernel Using Orthogonal Polynomial Method

This paper deals with the solution of a third kind mixed integro-differential equation (MIDE) in displacement type in space L 2 − 1 , 1 × C 0 , T , T < 1 . The singular kernel is modified to take a logarithmic form, while the kernels of time are continuous and positive functions. Using the separation of variables technique, we have a system of Fredholm integral equations (FIEs) that can be transformed into an algebraic system after using orthogonal polynomials. In all the previous researchers’ works, the time periods were divided, and the mixed equation transformed into an algebraic system of FIEs. While when using the separation method, we are able to obtain FIE with time coefficients, and these functions are described as an integral operator in time. Thus, we can study the behavior of the solution with the time dimension in a broader and deeper than the previous one. Some examples are given and discussed to show the performance and efficiency of the proposed methods.

Performance Improvement for double-stator permanent magnet synchronous machine using analytical subdomain model

An analytical subdomain model is employed in this paper for predicting the magnetic field distributions in a three-phase double-stator permanent magnet synchronous machine (DS-PMSM) during open-circuit and onload conditions. Due to the stator cores are located in the outer and inner parts of the motor, the DS-PMSM construction is quite complex. The rotor magnets are positioned between these two stators. The stator inner radius, stator outer radius, slot opening, magnet arc, magnet thickness, inner and outer air-gap thickness and number of windings turns will directly influence the motor performance in DS-PMSM. The analytical subdomain model employed in this paper has a significant advantage as a rapid design tool since it is capable of precisely predicting the performance of DS-PMSM while requiring less computational effort. The analytical model was initially created using the separation of variables technique in four subdomains based on the Poisson’s and Laplace’s equations: inner air-gap, inner magnet, outer magnet and outer air-gap. Applying the appropriate boundary and interface conditions yields the field solutions in each subdomain. Besides, the fractional DS-PMSM with different number of slots between outer and inner stators to rotor poles can result in low cogging torque and non-overlapping winding configuration. The analytical results are validated by Finite Element Analysis (FEA). The slotted air-gap flux density, back-emf, and output torque have all been evaluated as electromagnetic performances. The results demonstrate that the suggested analytical model is capable of accurately predicting the DS-PMSM performance.

Exotic vector freak waves in the nonlocal nonlinear Schrödinger equation

In this paper, general higher-order freak wave solutions of the nonlocal nonlinear Schrödinger equation (NLSE) with parity-time symmetric can be calculated theoretically via a Darboux transformation and a separation of variable technique. This family of solutions are given in separation of variables form. Moreover, in order to understand these obtained solutions better, the main characteristics of two lowest freak wave solutions are discussed clearly and conveniently. They show that the dynamics of these solutions exhibits rich patterns, most of which have no counterparts in the corresponding local equation.

A revised mathematical solution to the Penne bioheat equation for deep rooted tissues using the variable separation method

The temperature distribution of tissues under the skin surface was obtained using Penne’s bio-heat equation and the separation of variables technique. The resulting solution indicated a slight temperature difference between the recorded skin temperature and the temperature of the tissue of interest (TOI), which is a direct indication of the TOI's temperature. According to the graphical analysis, temperature is proportionally related to time and tissue distance from the skin surface. As a result, the given analytic solution may be utilized to easily analyze deeprooted tissues while accounting for thermal properties.

Solution of the Hamilton – Jacobi Equation in a Central Potential Using the Separation of Variables Method with Staeckel Boundary Conditions

Abstract: This manuscript aims at solving Hamilton-Jacobi equation in a central potential using the separation of variables technique with Staeckel boundary conditions. Our results show that the Hamilton – Jacobi variables can be completely separated, which agrees with other results employing different methods. Keywords: Lagrangian mechanics, Hamilton-Jacobi, Staeckel boundary conditions, Staeckel matrix, Staeckel vector, Hamilton's characteristic function, Hamilton's principal function.

Fractional Fourier Series with Separation of Variables Technique and Its Application on Fractional differential Equations

When using some classical methods, such us separation of variables; it is impossible to find a general solution for some differential equations. Therefore, we suggest adding conformable fractional Fourier series to get a new technique to solve fractional Benjamin Bana Mahony and Heat Equations. Furtheremore, we give new numerical approximation for functions using mathematica coding called conformable fractional Fourier series approximation

ANALYTICAL APPROACH TO THE PENNE’S BIOHEAT EQUATION FOR THE EVALUATION OF TEMPERATURE FOR DEEP SEATED TISSUES

In clinical practice, body temperature is an acclaimed indicator for diagnosis and treatments amongst other follow - ups. This work studied the temperature distribution in deep-seated tissues using Penne’s bio-heat equation (PBE). The analytical solution of the PBE was formulated using the separation of variable technique. The resulting solution gives an expression for tissue temperature in organs that are located inside the body. This revealed that the tissue temperature is subject to changes in tissue thickness, the distance of tissue from the skin surface and other thermal properties of tissues under consideration.

Modeling of cooling and heat conduction in permanent mold casting process

Motivated by the need to understand the heat transfer process in permanent mold casting, the heat conduction problem in the casting and the mold is modeled as transient one-dimensional heat flow in a double-layer cylinder with radial interfacial heat flux to emulate the cooling process of the casting. The cylinder is cooled down by dissipating heat to surrounding by convection. Green's function method, which obtained by separation of variables technique, is used to obtain a closed form solution of temperature distribution. The results of the derived analytical expressions are verified with numerical results of finite-element analysisand the published experimental results. The simulations are performed for different casting/mold materials with diversified thermo-physical properties to figure out the relationship between those properties and heat transfer process. The analytical results are justified by their good agreement with both of numerical and experimental results and its time efficiency in computation which offers advantages in potential real-time application to casting process monitoring. The mean absolute percentage error between the obtained temperatures using the developed analytical model and the measured ones was 14.5% and 2.4% for the casting and mold, respectively.

Evaluation of analytical solution of advection diffusion equation in three dimensions

AbstractThree‐dimensional advection–diffusion equation with steady state was evaluated from continuous point source taking the vertical and crosswind eddy diffusivities as power law of vertical height and downwind distance with constant wind speed to get the concentration in three dimensions. Separation of variables technique and Hankel transform were used to calculate this equation. The proposed analytical solution was compared with diffusion experiment of radioactive Iodine‐135 (I135) in unstable condition at Inshas, Cairo, Egypt. A good agreement achieved between calculated and observed concentration support our proposed model.

The Graetz-Nusselt problem for the curved channel using spectral collocation method

This paper investigates the Graetz-Nusselt problem for a curved channel. The solution of the energy equation for constant wall temperature boundary condition is developed via the separation of variable technique. The associated eigenvalue problem is handled numerically with the help of the spectral collocation method, and the Simpson’s 1/3 rule is employed to compute the coefficient of the solution series. The impact of curvature on temperature profile, mean temperature, and Nusselt numbers at both upper and lower walls of the curved channel is presented through tables and graphs and discussed in detail. The analysis reveals an attenuation in the net heat transfer rate through the channel with raising the curvature parameter. The local Nusselt number at the upper wall of the curved channel decreases while at the lower wall it follows an increasing trend with enhancing the channel curvature. In addition, the net heat transfer rate through the curved channel is significantly higher than its counterpart for a straight channel. It is believed that the availability of such a solution will be an important contribution in the field of biomedical sciences, engineering sciences and development of many industrial types of equipment.

Axisymmetric vibrations of temperature-dependent functionally graded moderately thick circular plates with two-dimensional material and temperature distribution

In the present paper, free vibration analysis of bi-directional functionally graded circular plates subjected to two-dimensional temperature variation has been presented on the basis of first-order shear deformation theory. The mechanical properties of the plate material are assumed to be temperature-dependent and graded in thickness as well as radial direction. Using the thermal boundary conditions of the plate, the exact solution of two-dimensional heat conduction equation has been obtained by separation of variables technique. The thermoelastic equilibrium equations and equations of motion for such plate have been derived from an energy-based Hamilton’s principle. The generalized differential quadrature method has been applied to compute the numerical values for thermally induced displacements and natural frequencies. MATLAB has been used to obtain these values for clamped and simply supported plates. The influence of thermal environment together with various plate parameters such as power-law index, heterogeneity parameter and density parameter on the natural frequencies has been investigated for the first three modes. The results obtained herein for some special cases are in good agreement with those obtained by other numerical methods. Three-dimensional mode shapes for specified plates have been illustrated.

Solution of the Hamilton – Jacobi Equations in an Electromagnetic Field Using Separation of Variables Method – Staeckel Boundary Conditions

This manuscript aims to resolve the Hamilton-Jacobi equations in an electromagnetic field by two methods. The first uses the separation of variables technique with Staeckel boundary conditions, whereas the second uses the Newtonian formalism to solve the same example. Our results demonstrate that the Hamilton-Jacobi variables can be completely detached by using separation of variables technique with Staeckel boundary conditions that correspond to other results using Newtonian formalism.

Inlet swirl decay and mixing in a laminar micro-pipe flow with wall slip

In this work, the decaying laminar inlet swirl flow in a straight circular micro-pipe with wall slip is solved analytically and the solution is verified numerically. Based on a fully developed parabolic axial velocity profile, the swirl velocity equation is solved by the separation of variables technique. The solution is expressed as a function of the flow Reynolds number, the axial distance within the circular micro-pipe from the inlet, the wall slip, and the inlet swirl velocity profile. The effects of the parameters on the swirl velocity distribution and the swirl decay are analyzed along the flow. The addition of swirling velocity to the flow of a fluid in a pipe is of great importance in the enhancement of transport characteristics. The current results offer analytical equations to estimate the swirl velocity distribution with slip at the walls for the design of swirl flow devices. Furthermore, to quantify mixing, the change in the average distance traveled by fluid particles from the inlet in a swirl flow is compared with the average distance traveled by the fluid particles in the case of no swirl. A clear enhancement of the average distance traveled is obtained for flows with the interaction of both swirl and slip effects. In our opinion, the present work is useful to researchers looking for the enhancement of transport characteristics in circular micro-pipes.

Portfolio Selection Strategies with Return Clause in a DC Pension Fund

This paper solves the problem faced by a pension fund manager in determining the optimal selection strategies involving four different assets comprising of one risk free asset and three risky assets whose prices are modelled by geometric Brownian motion. Also, a clause mandating the fund managers to return the accumulations with predetermined interest to members who lost their life during the accumulation period is considered. A stochastic optimal control model is formulated comprising of member’s monthly contributions, invested funds and the returned contributions. Also, an optimization problem from the extended Hamilton Jacobi Bellman (HJB) equation is established using the game theoretic approach. The explicit solutions of the optimal selection strategies and the efficient frontier are obtained by solving the extended HJB equation using the mean variance utility and separation of variable technique. Furthermore, a sensitivity analysis of the effect of some parameters on the optimal selection strategies is carried out numerically.

On dynamic pull‐in instability of electrostatically actuated multilayer nanoresonators: A semi‐analytical solution

AbstractBased on the nonlocal Euler–Bernoulli beam theory, a theoretical approach is developed to investigate the effects of small scale and intermolecular force on the dynamic pull‐in behavior of electrostatically actuated nanoresonators. To this purpose, nanoresonators are modeled as multilayer beams with rectangular cross‐sections and fixed‐fixed and fixed‐free end conditions which are embedded in an elastic medium containing Winkler and Pasternak elastic foundations. Also, the effects of nonlocal parameter, fringing field due to the finite width of beams, Casimir or van der Waals intermolecular forces, nonlinear term induced by mid‐plane stretching and Winkler and Pasternak elastic foundations are considered in the mathematical modeling of pull‐in behavior. Finally, a hybrid semi‐analytical method is proposed to solve the nonlinear partial differential equation in two steps using the separation of variable technique. First, the Galerkin method is used to reduce a nonlinear ordinary differential equation with respect to time variable. Next, the parameter expansion method is applied to solve ensuing equation and obtain the time history response of transverse deflection. As case studies, the numerical results including curves of voltage‐normalized frequency, time history of normalized maximum deflection and phase portrait of two six‐layer cantilever and fully fixed beams are presented and discussed in detail. Findings indicate that the effect of small scale on the pull‐in voltages is more significant for both boundary conditions, in comparison with the intermolecular force. Also, it is seen that the pull‐in voltages of cantilever beams become larger by considering the nonlocal effect, contrary to the case of fully fixed beams.

Water wave diffraction by a surface-piercing floating compound porous cylinder in finite depth

ABSTRACTThe present study deals with the hydrodynamic force due to water wave interaction with a floating compound porous cylinder in finite depth. The bottom and top of the lower cylinder and the side-wall of the upper cylinder (r=a) are impermeable. The problem is analysed under the assumption of small amplitude water wave theory. The sea-bed is considered to be flat. The velocity potentials are analytically derived in each fluid region based on separation of variables technique. The velocity potentials satisfy appropriate free surface condition, bottom boundary condition, matching conditions and Sommerfeld radiation condition at infinity. By using matching conditions along the boundaries of the regions, a system of linear equations for the unknown coefficients is derived and solved. A set of values of hydrodynamic force and wave run-up are obtained for different radii, different drafts and different porosity of the cylinder. It is observed that change in values in radii, draft and porosity have a significant effect on the hydrodynamic loads and wave run-up. We also analyse the hydrodynamic force due to different gaps between the cylinder and the sea-bed. The behaviour of hydrodynamic load is observed to be steady in the lower frequency. However, fluctuations are observed due to the resonance situation in the neighbourhood of a specific frequency. The hydrodynamic force almost vanishes at higher frequencies. Comparison is carried out with a floating compound cylinder, i.e. porous wall not considered in the cylinder. Useful agreement is observed from this comparison. It is expected that our result will be helpful in designing appropriate marine structures.

Hydrodynamical study of flow in a permeable channel: Application to flat plate dialyzer

This article investigates the problem of creeping motion of a Newtonian fluid in a permeable slit. The seepage velocity at the slit walls is assumed to obey the Darcy's law whereas the tangential velocity is taken in accordance with the Beaver's and Joseph's slip boundary condition. Exact solution of the hydrodynamical equations is obtained using the separation of variables technique. Expressions describing various quantities of interests are also derived. The influence of slip parameter is observed on the velocity and pressure field. The obtained results are applied to the physiological problem of blood flow in a flat plate dialyzer. Using the available data in the literature, theoretical values of filtration coefficient and the mean pressure drop in a flat plate dialyzer are computed and found to be in good agreement with the corresponding available experimental values.

Analytical Method for Magnetic Field and Electromagnetic Performances in Switched Reluctance Machines

This paper presents a new analytical method for the computation of magnetic field distributions and electromagnetic performances in switched reluctance machines (SRM). The proposed model is sufficiently general to be used with any number of stator slots and rotor poles with electrically excited. Due to the inherent nonlinear characteristics of SRM, the conventional analytic model has a limited accuracy which does not take into account the saturation. In order to calculate the accuracy and consider the local magnetic saturation on the stator and rotor teeth, the proposed method is based on the resolution of 2-D Laplace’s and Poisson’s equations in polar coordinates by the separation of variables technique. The solution region is divided into three types of regular subdomains, viz. air gap, stator slots and rotor slots. Magnetic field distributions, flux linkage, self-inductance, mutual inductance and static electromagnetic torque computed with the proposed analytical method are validated through finite-element method.