Search results for: numerical solution of the Navier-Stokes equations

Authors: Lev Idels, Arcady Ponosov

Abstract:

Stochastic fractional differential equations have recently attracted considerable attention, as they have been used to model real-world processes, which are subject to natural memory effects and measurement uncertainties. Compared to conventional hereditary differential equations, one of the advantages of fractional differential equations is related to more realistic geometric properties of their trajectories that do not intersect in the phase space. In this report, a Peano-like existence theorem for nonlinear stochastic fractional differential equations is proven under very general hypotheses. Several specific classes of equations are checked to satisfy these hypotheses, including delay equations driven by the fractional Brownian motion, stochastic fractional neutral equations and many others.

Keywords: delay equations, operator methods, stochastic noise, weak solutions

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Authors: Salah Alrabeei, Mohammad Yousuf

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The goal of option pricing theory is to help the investors to manage their money, enhance returns and control their financial future by theoretically valuing their options. However, most of the option pricing models have no analytical solution. Furthermore, not all the numerical methods are efficient to solve these models because they have nonsmoothing payoffs or discontinuous derivatives at the exercise price. In this paper, we solve the American option under jump diffusion models by using efficient time-dependent numerical methods. several techniques are integrated to reduced the overcome the computational complexity. Fast Fourier Transform (FFT) algorithm is used as a matrix-vector multiplication solver, which reduces the complexity from O(M2) into O(M logM). Partial fraction decomposition technique is applied to rational approximation schemes to overcome the complexity of inverting polynomial of matrices. The proposed method is easy to implement on serial or parallel versions. Numerical results are presented to prove the accuracy and efficiency of the proposed method.

Keywords: integral differential equations, jump–diffusion model, American options, rational approximation

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Authors: Mohammad Reza Shahnazari, Reza Kazemi, Ali Saberi

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Most of the engineering problems are in nonlinear forms. Nonlinear boundary layer problems defined in infinite intervals contain specific complexities, especially in boundary layer condition conformance. As an example of these nonlinear complex problems, the well-known Blasius equation can be mentioned, which itself is one of the classic boundary layer problems. No analytical solution has been proposed yet for the Blasius equation due to its complexity. In this paper, an analytical method, namely the Kourosh method, based on the singularity perturbation method and the Liao homotopy analysis is utilized to solve the Blasius problem. In this method, an inner solution is developed in the [0,1] interval to expedite the solution convergence. The magnitude of the f ˝(0), as an essential quantity for determining the physical parameters, is directly calculated from the solution of the boundary condition problem. The advantages of this solution are that it does not need any numerical solution, it has a closed form and that its validation is shown in the entire [0,∞] interval. Furthermore, all of the desirable parameters could be extracted through a series of simple analytical operations from the final solution. This solution also satisfies the continuity conditions, which is one of the main contributions of this paper in comparison with most of the other proposed analytical solutions available in the literature. Comparison with numerical solutions reveals that the proposed method is highly accurate and convenient for application.

Keywords: Blasius equation, boundary layer, Kourosh method, analytical solution

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Authors: Raphael de Oliveira Garcia, Samuel Rocha de Oliveira

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We have developed a new computer program in Fortran 90, in order to obtain numerical solutions of a system of Relativistic Magnetohydrodynamics partial differential equations with predetermined gravitation (GRMHD), capable of simulating the formation of relativistic jets from the accretion disk of matter up to his ejection. Initially we carried out a study on numerical methods of unidimensional Finite Volume, namely Lax-Friedrichs, Lax-Wendroff, Nessyahu-Tadmor method and Godunov methods dependent on Riemann problems, applied to equations Euler in order to verify their main features and make comparisons among those methods. It was then implemented the method of Finite Volume Centered of Nessyahu-Tadmor, a numerical schemes that has a formulation free and without dimensional separation of Riemann problem solvers, even in two or more spatial dimensions, at this point, already applied in equations GRMHD. Finally, the Nessyahu-Tadmor method was possible to obtain stable numerical solutions - without spurious oscillations or excessive dissipation - from the magnetized accretion disk process in rotation with respect to a central black hole (BH) Schwarzschild and immersed in a magnetosphere, for the ejection of matter in the form of jet over a distance of fourteen times the radius of the BH, a record in terms of astrophysical simulation of this kind. Also in our simulations, we managed to get substructures jets. A great advantage obtained was that, with the our code, we got simulate GRMHD equations in a simple personal computer.

Keywords: finite volume methods, central schemes, fortran 90, relativistic astrophysics, jet

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Authors: Gleda Kutrolli, Maksi Kutrolli, Etjon Meco

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SARS-CoV-2 virus is currently one of the most infectious pathogens for humans. It started in China at the end of 2019 and now it is spread in all over the world. The origin and diffusion of the SARS-CoV-2 epidemic, is analysed based on the discussion of viral phylogeny theory. With the aim of understanding the spread of infection in the affected countries, it is crucial to modelize the spread of the virus and simulate its activity. In this paper, the prediction of coronavirus outbreak is done by using SIR model without vital dynamics, applying different numerical technique solving ordinary differential equations (ODEs). We find out that ABM and MRT methods perform better than other techniques and that the activity of the virus will decrease in April but it never cease (for some time the activity will remain low) and the next cycle will start in the middle July 2020 for Norway and Denmark, and October 2020 for Sweden, and September for Finland.

Keywords: forecasting, ordinary differential equations, SARS-COV-2 epidemic, SIR model

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Authors: Gulgassyl Nugmanova, Zhanat Zhunussova, Kuralay Yesmakhanova, Galya Mamyrbekova, Ratbay Myrzakulov

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In this paper, we consider some integrable Heisenberg Ferromagnet Equations with self-consistent potentials. We study their Lax representations. In particular we derive their equivalent counterparts in the form of nonlinear Schr\"odinger type equations. We present the integrable reductions of the Heisenberg Ferromagnet Equations with self-consistent potentials. These integrable Heisenberg Ferromagnet Equations with self-consistent potentials describe nonlinear waves in ferromagnets with some additional physical fields.

Keywords: Heisenberg Ferromagnet equations, soliton equations, equivalence, Lax representation

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Authors: Mohammadreza Akbari, Sara Akbari, Davood Domiri Ganji, Pooya Solimani, Reza Khalili

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As all experts know most of engineering system behavior in practical are nonlinear process (especially heat, fluid and mass, etc.) and analytical solving (no numeric) these problems are difficult, complex and sometimes impossible like (fluids and gas wave, these problems can't solve with numeric method, because of no have boundary condition) accordingly in this symposium we are going to exposure a innovative approach which we have named it Akbari-Ganji's Method or AGM in engineering, that can solve sets of coupled nonlinear differential equations (ODE, PDE) with high accuracy and simple solution and so this issue will be emerged after comparing the achieved solutions by Numerical method (Runge-Kutte 4th) and so compare to other methods such as HPM, ADM,… and exact solutions. Eventually, AGM method will be proved that could be created huge evolution for researchers, professors and students (engineering and basic science) in whole over the world, because of AGM coding system, so by using this software we can analytically solve all complicated linear and nonlinear differential equations, with help of that there is no difficulty for solving nonlinear differential equations(ODE and PDE). In this paper, we investigate and solve 4 types of the nonlinear differential equation with AGM method : 1-Heat and fluid, 2-Unsteady state of nonlinear partial differential, 3-Coupled nonlinear partial differential in wave equation, and 4-Nonlinear integro-differential equation.

Keywords: new method AGM, sets of coupled nonlinear equations at engineering field, waves equations, integro-differential, fluid and thermal

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Authors: Peña A. Roland R., Lozano P. Jean P.

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The waterflooding process is an enhanced oil recovery (EOR) method that appears tremendously successful. This paper shows the importance of the role of the numerical modelling of waterflooding and how to provide a better description of the fluid flow during this process. The mathematical model is based on the mass conservation equations for the oil and water phases. Rock compressibility and capillary pressure equations are coupled to the mathematical model. For discretizing and linearizing the partial differential equations, we used the Finite Volume technique and the Newton-Raphson method, respectively. The results of three scenarios for waterflooding in porous media are shown. The first scenario was estimating the water saturation in the media without rock compressibility and without capillary pressure. The second scenario was estimating the front of the water considering the rock compressibility and capillary pressure. The third case is to compare different fronts of water saturation for three fluids viscosity ratios without and with rock compressibility and without and with capillary pressure. Results of the simulation indicate that the rock compressibility and the capillary pressure produce changes in the pressure profile and saturation profile during the displacement of the oil for the water.

Keywords: capillary pressure, numerical simulation, rock compressibility, two-phase flow

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Authors: Ahlem Mougaida, Hedi Bel Hadj Salah

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Several methods are suggested to improve the numerical integration in Galerkin weak form for Meshless methods. In fact, integrating without taking into account the characteristics of the shape functions reproduced by Meshless methods (rational functions, with compact support etc.), causes a large integration error that influences the PDE’s approximate solution. Comparisons between different methods of numerical integration for rational functions are discussed and compared. The algorithms are implemented in Matlab. Finally, numerical results were presented to prove the efficiency of our algorithms in improving results.

Keywords: adaptive methods, meshless, numerical integration, rational quadrature

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Authors: Jiangxia Han, Liang Xue, Keda Chen

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Despite the significant progress over the last decades in reservoir simulation using numerical discretization, meshing is complex. Moreover, the high degree of freedom of the space-time flow field makes the solution process very time-consuming. Therefore, we present Physics-Informed Convolutional Neural Networks(PICNN) as a hybrid scientific theory and data method for reservoir modeling. Besides labeled data, the model is driven by the scientific theories of the underlying problem, such as governing equations, boundary conditions, and initial conditions. PICNN integrates governing equations and boundary conditions into the network architecture in the form of a customized convolution kernel. The loss function is composed of data matching, initial conditions, and other measurable prior knowledge. By customizing the convolution kernel and minimizing the loss function, the neural network parameters not only fit the data but also honor the governing equation. The PICNN provides a methodology to model and history-match flow and transport problems in porous media. Numerical results demonstrate that the proposed PICNN can provide an accurate physical solution from a limited dataset. We show how this method can be applied in the context of a forward simulation for continuous problems. Furthermore, several complex scenarios are tested, including the existence of data noise, different work schedules, and different good patterns.

Keywords: convolutional neural networks, deep learning, flow and transport in porous media, physics-informed neural networks, reservoir simulation

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Authors: A. Giniatoulline

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A model of the mathematical fluid dynamics which describes the motion of a three-dimensional viscous rotating fluid in a homogeneous gravitational field with the consideration of the salinity and heat transfer is considered in a vertical finite layer. The model is a generalization of the linearized Navier-Stokes system with the addition of the Coriolis parameter and the equations for changeable density, salinity, and heat transfer. An explicit solution is constructed and the proof of the existence and uniqueness theorems is given. The localization and the structure of the spectrum of inner waves is also investigated. The results may be used, in particular, for constructing stable numerical algorithms for solutions of the considered models of fluid dynamics of the Atmosphere and the Ocean.

Keywords: Fourier transform, generalized solutions, Navier-Stokes equations, stratified fluid

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Authors: Mohammad R. Jalali

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Movement of liquid with a free surface in a container is known as slosh. For instance, slosh occurs when water in a closed tank is set in motion by a free surface displacement, or when liquid natural gas in a container is vibrated by an external driving force, such as an earthquake or movement induced by transport. Slosh is also derived from resonant switching of a natural basin. During sloshing, different types of motion are produced by energy exchange between the liquid and its container. In present study, a numerical model is developed to simulate the nonlinear even harmonic oscillations of free surface sloshing of an initial disturbance to the free surface of a liquid in a closed square basin. The response of the liquid free surface is affected by amplitude and motion frequencies of its container; therefore, sloshing involves complex fluid-structure interactions. In the present study, nonlinear interaction of free surface sloshing of an initial Gaussian hump with its uneven container is predicted numerically. For this purpose, Green-Naghdi (GN) equations are applied as governing equation of fluid field to produce nonlinear second-order and higher-order wave interactions. These equations reduce the dimensions from three to two, yielding equations that can be solved efficiently. The GN approach assumes a particular flow kinematic structure in the vertical direction for shallow and deep-water problems. The fluid velocity profile is finite sum of coefficients depending on space and time multiplied by a weighting function. It should be noted that in GN theory, the flow is rotational. In this study, GN numerical simulations of initial Gaussian hump are compared with Fourier series semi-analytical solutions of the linearized shallow water equations. The comparison reveals that satisfactory agreement exists between the numerical simulation and the analytical solution of the overall free surface sloshing patterns. The resonant free surface motions driven by an initial Gaussian disturbance are obtained by Fast Fourier Transform (FFT) of the free surface elevation time history components. Numerically predicted velocity vectors and magnitude contours for the free surface patterns indicate that interaction of Gaussian hump with its container has localized effect. The result of this sloshing is applicable to the design of stable liquefied oil containers in tankers and offshore platforms.

Keywords: fluid-structure interactions, free surface sloshing, Gaussian hump, Green-Naghdi equations, numerical predictions

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Authors: Ayan Chakraborty, BV. Rathish Kumar

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Off-late many models in viscoelasticity, signal processing or anomalous diffusion equations are formulated in fractional calculus. Tempered fractional calculus is the generalization of fractional calculus and in the last few years several important partial differential equations occurring in the different field of science have been reconsidered in this term like diffusion wave equations, Schr$\ddot{o}$dinger equation and so on. In the present paper, a time-dependent tempered fractional diffusion equation of order $\gamma \in (0,1)$ with forcing function is considered. Existence, uniqueness, stability, and regularity of the solution has been proved. Crank-Nicolson discretization is used in the time direction. B spline finite element approximation is implemented. Generally, B-splines basis are useful for representing the geometry of a finite element model, interfacing a finite element analysis program. By utilizing this technique a priori space-time estimate in finite element analysis has been derived and we proved that the convergent order is $\mathcal{O}(h²+T²)$ where $h$ is the space step size and $T$ is the time. A couple of numerical examples have been presented to confirm the accuracy of theoretical results. Finally, we conclude that the studied method is useful for solving tempered fractional diffusion equations.

Keywords: B-spline finite element, error estimates, Gronwall's lemma, stability, tempered fractional

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Authors: Ayesha Sohail, Khadija Maqbool, Anila Asif, Haroon Ahmad

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The field of tissue engineering is an active area of research. Bone tissue engineering helps to resolve the clinical problems of critical size and non-healing defects by the creation of man-made bone tissue. We will design and validate an efficient numerical model, which will simulate the effective diffusivity in bone tissue engineering. Our numerical model will be based on the finite element analysis of the diffusion-reaction equations. It will have the ability to optimize the diffusivity, even at multi-scale, with the variation of time. It will also have a special feature, with which we will not only be able to predict the oxygen, glucose and cell density dynamics, more accurately, but will also sort the issues arising due to anisotropy. We will fix these problems with the help of modifying the governing equations, by selecting appropriate spatio-temporal finite element schemes, by adaptive grid refinement strategy and by transient analysis.

Keywords: scaffolds, porosity, diffusion, transient analysis

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Authors: Kwan-u Kim, Jin-Sim, Ryong-Jin Jang, Sung-Duk Kim

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106 years have passed since a great number of physicists explained astronomical and physical phenomena by solving geodesic equations in Schwarzschild metric. However, when solving the geodesic equations in Schwarzschild metric, they did not solve correctly one branch of the component of space among spatial and temporal components of four-dimensional force and did not come up with physical laws correctly by means of physical analysis from the results obtained by solving the geodesic equations. In addition to it, they did not treat the astronomical and physical phenomena in a physical way based on the exact physical laws obtained from the solution of the geodesic equations in Schwarzschild metric, so some former scholars mentioned that Einstein’s theoretical basis of general theory of relativity was obscure and incorrect, but they have not given a correct physical solution to the problems. Furthermore, because general theory of relativity has not given a quantitative solution to the obscure and incorrect problems, generalization of gravitational theory although the former scholars thought of it and tried to do it. In order to solve the problems it is necessary to explore theobscure and incorrect problems in general theory of relativity based on the physical laws and to find out the methodology of solving the problems. Therefore, first of all, as the first sarytep for achieving the purpose, the correct solution of the geodesic equation in Schwarzschild metric has been presented. Next, the correct physical laws found by making a physical analysis of the results has been presented, the obscure and incorrect problems have been showed and an analysis of them has been made on the basis of the physical laws. In addition, the experimental verification of the physical laws found by us has been made.

Keywords: equivalence principle, general relativity, geometrodynamics, Schwarzschild

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Authors: Somveer Singh, Vineet Kumar Singh

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We present a new model based on viscoelasticity for the Non-Newtonian fluids.We use a matrix formulated algorithm to approximate solutions of a class of partial integro-differential equations with the given initial and boundary conditions. Some numerical results are presented to simplify application of operational matrix formulation and reduce the computational cost. Convergence analysis, error estimation and numerical stability of the method are also investigated. Finally, some test examples are given to demonstrate accuracy and efficiency of the proposed method.

Keywords: Legendre Wavelets, operational matrices, partial integro-differential equation, viscoelasticity

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Authors: Fernando Maass, Pablo Martin, Jorge Olivares

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Fractional derivatives have become very important in several areas of Engineering, however, the solutions of simple differential equations are not known. Here we are considering the simplest first order nonhomogeneous differential equations with Bessel regular functions of the first kind, in this way the solutions have been found which are hypergeometric solutions for any fractional derivative of order α, where α is rational number α=m/p, between zero and one. The way to find this result is by using Laplace transform and the Caputo definitions of fractional derivatives. This method is for values longer than one. However for α entire number the hypergeometric functions are Kumer type, no integer values of alpha, the hypergeometric function is more complicated is type ₂F₃(a,b,c, t2/2). The argument of the hypergeometric changes sign when we go from the regular Bessel functions to the modified Bessel functions of the first kind, however it integer seems that using precise values of α and considering no integers values of α, a solution can be obtained in terms of two hypergeometric functions. Further research is required for future papers in order to obtain the general solution for any rational value of α.

Keywords: Caputo, fractional calculation, hypergeometric, linear differential equations

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Authors: Fakhreddin Abedi, Wah June Leong

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Exponential stability of stochastic differential equations with non trivial solutions is provided in terms of Lyapunov functions. The main result of this paper establishes that, under certain hypotheses for the dynamics f(.) and g(.), practical exponential stability in probability at the small neighborhood of the origin is equivalent to the existence of an appropriate Lyapunov function. Indeed, we establish exponential stability of stochastic differential equation when almost all the state trajectories are bounded and approach a sufficiently small neighborhood of the origin. We derive sufficient conditions for exponential stability of stochastic differential equations. Finally, we give a numerical example illustrating our results.

Keywords: exponential stability in probability, stochastic differential equations, Lyapunov technique, Ito's formula

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Authors: Mohammed Ali Hjaji, Magdi Mohareb

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This paper develops the exact solutions for coupled flexural-lateral-torsional static response of thin-walled asymmetric open members subjected to general loading. Using the principle of stationary total potential energy, the governing differential equations of equilibrium are formulated as well as the associated boundary conditions. The formulation is based on a generalized Timoshenko-Vlasov beam theory and accounts for the effects of shear deformation due to bending and warping, and captures the effects of flexural–torsional coupling due to cross-section asymmetry. Closed-form solutions are developed for cantilever and simply supported beams under various forces. In order to demonstrate the validity and the accuracy of this solution, numerical examples are presented and compared with well-established ABAQUS finite element solutions and other numerical results available in the literature. In addition, the results are compared against non-shear deformable beam theories in order to demonstrate the shear deformation effects.

Keywords: asymmetric cross-section, flexural-lateral-torsional response, Vlasov-Timoshenko beam theory, closed form solution

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Authors: Jessada Tariboon

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In this article, using distribution kernel, we study the nonlinear equations with n-dimensional telegraph operator iterated k-times.

Keywords: telegraph operator, elementary solution, distribution kernel, nonlinear equations

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Authors: Yilmaz Simsek

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In this paper, we study the Bernstein-type basis functions with their generating functions. We give various properties of these polynomials with the aid of their generating functions. These polynomials and generating functions have many valuable applications in mathematics, in probability, in statistics and also in mathematical physics. By using the Bernstein-Galerkin and the Bernstein-Petrov-Galerkin methods, we give some applications of the Bernstein-type polynomials for solving high even-order differential equations with their numerical computations. We also give Bezier-type curves related to the Bernstein-type basis functions. We investigate fundamental properties of these curves. These curves have many applications in mathematics, in computer geometric design and other related areas. Moreover, we simulate these polynomials with their plots for some selected numerical values.

Keywords: generating functions, Bernstein basis functions, Bernstein polynomials, Bezier curves, differential equations

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Authors: Salah Alrabeei, Mohammad Yousuf

Abstract:

The goal of option pricing theory is to help the investors to manage their money, enhance returns and control their financial future by theoretically valuing their options. Modeling option pricing by Black-School models with jumps guarantees to consider the market movement. However, only numerical methods can solve this model. Furthermore, not all the numerical methods are efficient to solve these models because they have nonsmoothing payoffs or discontinuous derivatives at the exercise price. In this paper, the exponential time differencing (ETD) method is applied for solving partial integrodifferential equations arising in pricing European options under Merton’s and Kou’s jump-diffusion models. Fast Fourier Transform (FFT) algorithm is used as a matrix-vector multiplication solver, which reduces the complexity from O(M2) into O(M logM). A partial fraction form of Pad`e schemes is used to overcome the complexity of inverting polynomial of matrices. These two tools guarantee to get efficient and accurate numerical solutions. We construct a parallel and easy to implement a version of the numerical scheme. Numerical experiments are given to show how fast and accurate is our scheme.

Keywords: Integral differential equations, , L-stable methods, pricing European options, Jump–diffusion model

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Authors: Mohsen Torabi, Kaili Zhang

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This work considers forced convection in a channel partially filled with porous media from local thermal non-equilibrium (LTNE) point of view. The channel is heated with constant heat flux from the lower side and is isolated on the top side. The wall heat flux is considered to be divided between the solid and fluid phases based on their temperature gradients and effective thermal conductivities. The general forms of the velocity and temperature fields are analytically obtained. To obtain the constant parameters for temperature equations, a numerical solution is considered. Using different thermophysical parameters, both velocity and temperature fields are comprehensively illustrated. Discussions regarding bifurcation phenomenon are provided. Since this geometry has not been considered yet, the present analysis is a useful addition to the literature on thermal performance of porous systems from LTNE perspective.

Keywords: local thermal non-equilibrium, forced convection, thermal bifurcation, porous-fluid interface, combined analytical-numerical solution

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Authors: Teng Li, Kamran Mohseni

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This paper presents an inviscid regularization technique that is capable of regularizing the shocks and sharp interfaces simultaneously in the shock-interface interaction simulations. The direct numerical simulation of flows involving shocks has been investigated for many years and a lot of numerical methods were developed to capture the shocks. However, most of these methods rely on the numerical dissipation to regularize the shocks. Moreover, in high Reynolds number flows, the nonlinear terms in hyperbolic Partial Differential Equations (PDE) dominates, constantly generating small scale features. This makes direct numerical simulation of shocks even harder. The same difficulty happens in two-phase flow with sharp interfaces where the nonlinear terms in the governing equations keep sharpening the interfaces to discontinuities. The main idea of the proposed technique is to average out the small scales that is below the resolution (observable scale) of the computational grid by filtering the convective velocity in the nonlinear terms in the governing PDE. This technique is named “observable method” and it results in a set of hyperbolic equations called observable equations, namely, observable Navier-Stokes or Euler equations. The observable method has been applied to the flow simulations involving shocks, turbulence, and two-phase flows, and the results are promising. In the current paper, the observable method is examined on the performance of regularizing shocks and interfaces at the same time in shock-interface interaction problems. Bubble-shock interactions and Richtmyer-Meshkov instability are particularly chosen to be studied. Observable Euler equations will be numerically solved with pseudo-spectral discretization in space and third order Total Variation Diminishing (TVD) Runge Kutta method in time. Results are presented and compared with existing publications. The interface acceleration and deformation and shock reflection are particularly examined.

Keywords: compressible flow simulation, inviscid regularization, Richtmyer-Meshkov instability, shock-bubble interactions.

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Authors: Riadh Zorgati, Thomas Triboulet

Abstract:

In quite diverse application areas such as astronomy, medical imaging, geophysics or nondestructive evaluation, many problems related to calibration, fitting or estimation of a large number of input parameters of a model from a small amount of output noisy data, can be cast as inverse problems. Due to noisy data corruption, insufficient data and model errors, most inverse problems are ill-posed in a Hadamard sense, i.e. existence, uniqueness and stability of the solution are not guaranteed. A wide class of inverse problems in physics relates to the Fredholm equation of the first kind. The ill-posedness of such inverse problem results, after discretization, in a very ill-conditioned linear system of equations, the condition number of the associated matrix can typically range from 109 to 1018. This condition number plays the role of an amplifier of uncertainties on data during inversion and then, renders the inverse problem difficult to handle numerically. Similar problems appear in other areas such as numerical optimization when using interior points algorithms for solving linear programs leads to face ill-conditioned systems of linear equations. Devising efficient solution approaches for such system of equations is therefore of great practical interest. Efficient iterative algorithms are proposed for solving a system of linear equations. The approach is based on a preconditioning of the initial matrix of the system with an approximation of a generalized inverse leading to a stochastic preconditioned matrix. This approach, valid for non-negative matrices, is first extended to hermitian, semi-definite positive matrices and then generalized to any complex rectangular matrices. The main results obtained are as follows: 1) We are able to build a generalized inverse of any complex rectangular matrix which satisfies the convergence condition requested in iterative algorithms for solving a system of linear equations. This completes the (short) list of generalized inverse having this property, after Kaczmarz and Cimmino matrices. Theoretical results on both the characterization of the type of generalized inverse obtained and the convergence are derived. 2) Thanks to its properties, this matrix can be efficiently used in different solving schemes as Richardson-Tanabe or preconditioned conjugate gradients. 3) By using Lp norms, we propose generalized Kaczmarz’s type matrices. We also show how Cimmino's matrix can be considered as a particular case consisting in choosing the Euclidian norm in an asymmetrical structure. 4) Regarding numerical results obtained on some pathological well-known test-cases (Hilbert, Nakasaka, …), some of the proposed algorithms are empirically shown to be more efficient on ill-conditioned problems and more robust to error propagation than the known classical techniques we have tested (Gauss, Moore-Penrose inverse, minimum residue, conjugate gradients, Kaczmarz, Cimmino). We end on a very early prospective application of our approach based on stochastic matrices aiming at computing some parameters (such as the extreme values, the mean, the variance, …) of the solution of a linear system prior to its resolution. Such an approach, if it were to be efficient, would be a source of information on the solution of a system of linear equations.

Keywords: conditioning, generalized inverse, linear system, norms, stochastic matrix

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Authors: M. H. Kazemi, D. Salac

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A semi-implicit phase field method for droplet evolution is proposed. Using the phase field Cahn-Hilliard equation, we are able to track the interface in multiphase flow. The idea of a semi-implicit finite difference scheme is reviewed and employed to solve two nonlinear equations, including the Navier-Stokes and the Cahn-Hilliard equations. The use of a semi-implicit method allows us to have larger time steps compared to explicit schemes. The governing equations are coupled and then solved by a GMRES solver (generalized minimal residual method) using modified Gram-Schmidt orthogonalization. To show the validity of the method, we apply the method to the simulation of a rising droplet, a leaky dielectric drop and the coalescence of drops. The numerical solutions to the phase field model match well with existing solutions over a defined range of variables.

Keywords: coalescence, leaky dielectric, numerical method, phase field, rising droplet, semi-implicit method

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Authors: D. Mehdizadeh, M. Rahimian, M. Eskandari-Ghadi

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This article is concerned with the determination of the static interaction of a vertically loaded rigid circular disc embedded at the interface of a horizontal layer sandwiched in between two different transversely isotropic half-spaces called as tri-material full-space. The axes of symmetry of different regions are assumed to be normal to the horizontal interfaces and parallel to the movement direction. With the use of a potential function method, and by implementing Hankel integral transforms in the radial direction, the government partial differential equation for the solely scalar potential function is transformed to an ordinary 4th order differential equation, and the mixed boundary conditions are transformed into a pair of integral equations called dual integral equations, which can be reduced to a Fredholm integral equation of the second kind, which is solved analytically. Then, the displacements and stresses are given in the form of improper line integrals, which is due to inverse Hankel integral transforms. It is shown that the present solutions are in exact agreement with the existing solutions for a homogeneous full-space with transversely isotropic material. To confirm the accuracy of the numerical evaluation of the integrals involved, the numerical results are compared with the solutions exists for the homogeneous full-space. Then, some different cases with different degrees of material anisotropy are compared to portray the effect of degree of anisotropy.

Keywords: transversely isotropic, rigid disc, elasticity, dual integral equations, tri-material full-space

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Authors: W. K. Zahra, M. A. El-Beltagy, R. R. Elkhadrawy

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So many practical problems in science and technology developed over the past decays. For instance, the mathematical boundary layer theory or the approximation of solution for different problems described by differential equations. When such problems consider large or small parameters, they become increasingly complex and therefore require the use of asymptotic methods. In this work, we consider the singularly perturbed boundary value problems which contain very small parameters. Moreover, we will consider these perturbation parameters as random variables. We propose a numerical method to solve this kind of problems. The proposed method is based on an exponential spline, Shishkin mesh discretization, and polynomial chaos expansion. The polynomial chaos expansion is used to handle the randomness exist in the perturbation parameter. Furthermore, the Monte Carlo Simulations (MCS) are used to validate the solution and the accuracy of the proposed method. Numerical results are provided to show the applicability and efficiency of the proposed method, which maintains a very remarkable high accuracy and it is ε-uniform convergence of almost second order.

Keywords: singular perturbation problem, polynomial chaos expansion, Shishkin mesh, two small parameters, exponential spline

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Authors: Shubham Jaiswal

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In the present article, a drive is taken to compute the solution of spatial fractional order advection-dispersion equation having source/sink term with given initial and boundary conditions. The equation is converted to a system of ordinary differential equations using second-kind shifted Chebyshev polynomials, which have finally been solved using finite difference method. The striking feature of the article is the fast transportation of solute concentration as and when the system approaches fractional order from standard order for specified values of the parameters of the system.

Keywords: spatial fractional order advection-dispersion equation, second-kind shifted Chebyshev polynomial, collocation method, conservative system, non-conservative system

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Authors: Javad Jamali Khouei, Mohammadreza Khoshravan

Abstract:

Considering that application of functionally graded material is increasing in most industries, it seems necessary to present a methodology for designing optimal profile of structures such as plate under mechanical loading which is highly consumed in industries. Therefore, volume fraction variation profile of functionally graded circular plate which has been considered two-directional is optimized so that stress of structure is minimized. For this purpose, equilibrium equations of two-directional functionally graded circular plate are solved by applying semi analytical-numerical method under mechanical loading and support conditions. By solving equilibrium equations, deflections and stresses are obtained in terms of control variables of volume fraction variation profile. As a result, the problem formula can be defined as an optimization problem by aiming at minimization of critical von-mises stress under constraints of deflections, stress and a physical constraint relating to structure of material. Then, the related problem can be solved with help of one of the metaheuristic algorithms such as genetic algorithm. Results of optimization for the applied model under constraints and loadings and boundary conditions show that functionally graded plate should be graded only in radial direction and there is no need for volume fraction variation of the constituent particles in thickness direction. For validating results, optimal values of the obtained design variables are graphically evaluated.

Keywords: two-directional functionally graded material, single objective optimization, semi analytical-numerical solution, genetic algorithm, graphical solution with contour

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