Search results for: Jacobi method
18658 Hamilton-Jacobi Treatment of Damped Motion
Authors: Khaled I. Nawafleh
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In this work, we apply the method of Hamilton-Jacobi to obtain solutions of Hamiltonian systems in classical mechanics with two certain structures: the first structure plays a central role in the theory of time-dependent Hamiltonians, whilst the second is used to treat classical Hamiltonians, including dissipation terms. It is proved that the generalization of problems from the calculus of variation methods in the nonstationary case can be obtained naturally in Hamilton-Jacobi formalism. Then, another expression of geometry of the Hamilton Jacobi equation is retrieved for Hamiltonians with time-dependent and frictional terms. Both approaches shall be applied to many physical examples.Keywords: Hamilton-Jacobi, time dependent lagrangians, dissipative systems, variational principle
Procedia PDF Downloads 16218657 Convergence of Generalized Jacobi, Gauss-Seidel and Successive Overrelaxation Methods for Various Classes of Matrices
Authors: Manideepa Saha, Jahnavi Chakrabarty
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Generalized Jacobi (GJ) and Generalized Gauss-Seidel (GGS) methods are most effective than conventional Jacobi and Gauss-Seidel methods for solving linear system of equations. It is known that GJ and GGS methods converge for strictly diagonally dominant (SDD) and for M-matrices. In this paper, we study the convergence of GJ and GGS converge for symmetric positive definite (SPD) matrices, L-matrices and H-matrices. We introduce a generalization of successive overrelaxation (SOR) method for solving linear systems and discuss its convergence for the classes of SDD matrices, SPD matrices, M-matrices, L-matrices and for H-matrices. Advantages of generalized SOR method are established through numerical experiments over GJ, GGS, and SOR methods.Keywords: convergence, Gauss-Seidel, iterative method, Jacobi, SOR
Procedia PDF Downloads 18218656 Exact Solutions of K(N,N)-Type Equations Using Jacobi Elliptic Functions
Authors: Edamana Krishnan, Khalil Al-Ghafri
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In this paper, modified K(n,n) and K(n+1,n+1) equations have been solved using mapping methods which give a variety of solutions in terms of Jacobi elliptic functions. The solutions when m approaches 0 and 1, with m as the modulus of the JEFs have also been deduced. The role of constraint conditions has been discussed.Keywords: travelling wave solutions, solitary wave solutions, compactons, Jacobi elliptic functions, mapping methods
Procedia PDF Downloads 30118655 Numerical Solution of Porous Media Equation Using Jacobi Operational Matrix
Authors: Shubham Jaiswal
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During modeling of transport phenomena in porous media, many nonlinear partial differential equations (NPDEs) encountered which greatly described the convection, diffusion and reaction process. To solve such types of nonlinear problems, a reliable and efficient technique is needed. In this article, the numerical solution of NPDEs encountered in porous media is derived. Here Jacobi collocation method is used to solve the considered problems which convert the NPDEs in systems of nonlinear algebraic equations that can be solved using Newton-Raphson method. The numerical results of some illustrative examples are reported to show the efficiency and high accuracy of the proposed approach. The comparison of the numerical results with the existing analytical results already reported in the literature and the error analysis for each example exhibited through graphs and tables confirms the exponential convergence rate of the proposed method.Keywords: nonlinear porous media equation, shifted Jacobi polynomials, operational matrix, spectral collocation method
Procedia PDF Downloads 43018654 On the Algorithmic Iterative Solutions of Conjugate Gradient, Gauss-Seidel and Jacobi Methods for Solving Systems of Linear Equations
Authors: Hussaini Doko Ibrahim, Hamilton Cyprian Chinwenyi, Henrietta Nkem Ude
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In this paper, efforts were made to examine and compare the algorithmic iterative solutions of the conjugate gradient method as against other methods such as Gauss-Seidel and Jacobi approaches for solving systems of linear equations of the form Ax=b, where A is a real n×n symmetric and positive definite matrix. We performed algorithmic iterative steps and obtained analytical solutions of a typical 3×3 symmetric and positive definite matrix using the three methods described in this paper (Gauss-Seidel, Jacobi, and conjugate gradient methods), respectively. From the results obtained, we discovered that the conjugate gradient method converges faster to exact solutions in fewer iterative steps than the two other methods, which took many iterations, much time, and kept tending to the exact solutions.Keywords: conjugate gradient, linear equations, symmetric and positive definite matrix, gauss-seidel, Jacobi, algorithm
Procedia PDF Downloads 14118653 Numerical Solution of Space Fractional Order Linear/Nonlinear Reaction-Advection Diffusion Equation Using Jacobi Polynomial
Authors: Shubham Jaiswal
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During modelling of many physical problems and engineering processes, fractional calculus plays an important role. Those are greatly described by fractional differential equations (FDEs). So a reliable and efficient technique to solve such types of FDEs is needed. In this article, a numerical solution of a class of fractional differential equations namely space fractional order reaction-advection dispersion equations subject to initial and boundary conditions is derived. In the proposed approach shifted Jacobi polynomials are used to approximate the solutions together with shifted Jacobi operational matrix of fractional order and spectral collocation method. The main advantage of this approach is that it converts such problems in the systems of algebraic equations which are easier to be solved. The proposed approach is effective to solve the linear as well as non-linear FDEs. To show the reliability, validity and high accuracy of proposed approach, the numerical results of some illustrative examples are reported, which are compared with the existing analytical results already reported in the literature. The error analysis for each case exhibited through graphs and tables confirms the exponential convergence rate of the proposed method.Keywords: space fractional order linear/nonlinear reaction-advection diffusion equation, shifted Jacobi polynomials, operational matrix, collocation method, Caputo derivative
Procedia PDF Downloads 43518652 Exact Solutions of a Nonlinear Schrodinger Equation with Kerr Law Nonlinearity
Authors: Muna Alghabshi, Edmana Krishnan
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A nonlinear Schrodinger equation has been considered for solving by mapping methods in terms of Jacobi elliptic functions (JEFs). The equation under consideration has a linear evolution term, linear and nonlinear dispersion terms, the Kerr law nonlinearity term and three terms representing the contribution of meta materials. This equation which has applications in optical fibers is found to have soliton solutions, shock wave solutions, and singular wave solutions when the modulus of the JEFs approach 1 which is the infinite period limit. The equation with special values of the parameters has also been solved using the tanh method.Keywords: Jacobi elliptic function, mapping methods, nonlinear Schrodinger Equation, tanh method
Procedia PDF Downloads 30718651 A Study of Hamilton-Jacobi-Bellman Equation Systems Arising in Differential Game Models of Changing Society
Authors: Weihua Ruan, Kuan-Chou Chen
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This paper is concerned with a system of Hamilton-Jacobi-Bellman equations coupled with an autonomous dynamical system. The mathematical system arises in the differential game formulation of political economy models as an infinite-horizon continuous-time differential game with discounted instantaneous payoff rates and continuously and discretely varying state variables. The existence of a weak solution of the PDE system is proven and a computational scheme of approximate solution is developed for a class of such systems. A model of democratization is mathematically analyzed as an illustration of application.Keywords: Hamilton-Jacobi-Bellman equations, infinite-horizon differential games, continuous and discrete state variables, political-economy models
Procedia PDF Downloads 36718650 The Rayleigh Quotient for Structural Element Vibration Analysis with Finite Element Method
Authors: Falek Kamel
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Various approaches are usually used in the dynamic analysis of beams vibrating transversally. For this, numerical methods allowing the solving of the general eigenvalue problem are utilized. The equilibrium equations describe the movement resulting from the solution of a fourth-order differential equation. Our investigation is based on the finite element method. The findings of these investigations are the vibration frequencies obtained by the Jacobi method. Two types of the elementary mass matrix are considered, representing a uniform distribution of the mass along with the element and concentrated ones located at fixed points whose number is increased progressively separated by equal distances at each evaluation stage. The studied beams have different boundary constraints representing several classical situations. Comparisons are made for beams where the distributed mass is replaced by n concentrated masses. As expected, the first calculus stage is to obtain the lowest number of beam parts that gives a frequency comparable to that issued from the Rayleigh formula. The obtained values are then compared to theoretical results based on the assumptions of the Bernoulli-Euler theory. These steps are used for the second type of mass representation in the same manner.Keywords: structural elements, beams vibrating, dynamic analysis, finite element method, Jacobi method
Procedia PDF Downloads 15618649 Mapping Methods to Solve a Modified Korteweg de Vries Type Equation
Authors: E. V. Krishnan
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In this paper, we employ mapping methods to construct exact travelling wave solutions for a modified Korteweg-de Vries equation. We have derived periodic wave solutions in terms of Jacobi elliptic functions, kink solutions and singular wave solutions in terms of hyperbolic functions.Keywords: travelling wave solutions, Jacobi elliptic functions, solitary wave solutions, Korteweg-de Vries equation
Procedia PDF Downloads 32018648 Hawking Radiation of Grumiller Black
Authors: Sherwan Kher Alden Yakub Alsofy
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In this paper, we consider the relativistic Hamilton-Jacobi (HJ) equation and study the Hawking radiation (HR) of scalar particles from uncharged Grumiller black hole (GBH) which is affordable for testing in astrophysics. GBH is also known as Rindler modified Schwarzschild BH. Our aim is not only to investigate the effect of the Rindler parameter A on the Hawking temperature (TH ), but to examine whether there is any discrepancy between the computed horizon temperature and the standard TH as well. For this purpose, in addition to its naive coordinate system, we study on the three regular coordinate systems which are Painlev´-Gullstrand (PG), ingoing Eddington- Finkelstein (IEF) and Kruskal-Szekeres (KS) coordinates. In all coordinate systems, we calculate the tunneling probabilities of incoming and outgoing scalar particles from the event horizon by using the HJ equation. It has been shown in detail that the considered HJ method is concluded with the conventional TH in all these coordinate systems without giving rise to the famous factor- 2 problem. Furthermore, in the PG coordinates Parikh-Wilczek’s tunneling (PWT) method is employed in order to show how one can integrate the quantum gravity (QG) corrections to the semiclassical tunneling rate by including the effects of self-gravitation and back reaction. We then show how these corrections yield a modification in the TH.Keywords: ingoing Eddington, Finkelstein, coordinates Parikh-Wilczek’s, Hamilton-Jacobi equation
Procedia PDF Downloads 60618647 Quantization of Damped Systems Based on the Doubling of Degrees of Freedom
Authors: Khaled I. Nawafleh
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In this paper, it provide the canonical approach for studying dissipated oscillators based on the doubling of degrees of freedom. Clearly, expressions for Lagrangians of the elementary modes of the system are given, which ends with the familiar classical equations of motion for the dissipative oscillator. The equation for one variable is the time reversed of the motion of the second variable. it discuss in detail the extended Bateman Lagrangian specifically for a dual extended damped oscillator time-dependent. A Hamilton-Jacobi analysis showing the equivalence with the Lagrangian approach is also obtained. For that purpose, the techniques of separation of variables were applied, and the quantization process was achieved.Keywords: doubling of degrees of freedom, dissipated harmonic oscillator, Hamilton-Jacobi, time-dependent lagrangians, quantization
Procedia PDF Downloads 5918646 Inverse Scattering for a Second-Order Discrete System via Transmission Eigenvalues
Authors: Abdon Choque-Rivero
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The Jacobi system with the Dirichlet boundary condition is considered on a half-line lattice when the coefficients are real valued. The inverse problem of recovery of the coefficients from various data sets containing the so-called transmission eigenvalues is analyzed. The Marchenko method is utilized to solve the corresponding inverse problem.Keywords: inverse scattering, discrete system, transmission eigenvalues, Marchenko method
Procedia PDF Downloads 13618645 On Algebraic Structure of Improved Gauss-Seide Iteration
Authors: O. M. Bamigbola, A. A. Ibrahim
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Analysis of real life problems often results in linear systems of equations for which solutions are sought. The method to employ depends, to some extent, on the properties of the coefficient matrix. It is not always feasible to solve linear systems of equations by direct methods, as such the need to use an iterative method becomes imperative. Before an iterative method can be employed to solve a linear system of equations there must be a guaranty that the process of solution will converge. This guaranty, which must be determined a priori, involve the use of some criterion expressible in terms of the entries of the coefficient matrix. It is, therefore, logical that the convergence criterion should depend implicitly on the algebraic structure of such a method. However, in deference to this view is the practice of conducting convergence analysis for Gauss-Seidel iteration on a criterion formulated based on the algebraic structure of Jacobi iteration. To remedy this anomaly, the Gauss-Seidel iteration was studied for its algebraic structure and contrary to the usual assumption, it was discovered that some property of the iteration matrix of Gauss-Seidel method is only diagonally dominant in its first row while the other rows do not satisfy diagonal dominance. With the aid of this structure we herein fashion out an improved version of Gauss-Seidel iteration with the prospect of enhancing convergence and robustness of the method. A numerical section is included to demonstrate the validity of the theoretical results obtained for the improved Gauss-Seidel method.Keywords: linear algebraic system, Gauss-Seidel iteration, algebraic structure, convergence
Procedia PDF Downloads 45318644 Synchronization of Chaotic T-System via Optimal Control as an Adaptive Controller
Authors: Hossein Kheiri, Bashir Naderi, Mohamad Reza Niknam
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In this paper we study the optimal synchronization of chaotic T-system with complete uncertain parameter. Optimal control laws and parameter estimation rules are obtained by using Hamilton-Jacobi-Bellman (HJB) technique and Lyapunov stability theorem. The derived control laws are optimal adaptive control and make the states of drive and response systems asymptotically synchronized. Numerical simulation shows the effectiveness and feasibility of the proposed method.Keywords: Lyapunov stability, synchronization, chaos, optimal control, adaptive control
Procedia PDF Downloads 46818643 Analysis of Transverse Vibrations in Uniform Beams Subject to Different End Restraints
Authors: Falek Kamel
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Free vibration analysis of beams, based on the assumptions of Bernoulli-Euler theory, has been extensively studied. Many research works have focused on the study of transverse vibrations under the application of different boundary conditions where different theories have been applied. The stiffness and mass matrices considered are those obtained by assembling those resulting from the use of the finite element method. The Jacobi method has been used to solve the eigenvalue problem. These well-known concepts have been applied to the study of beams with constant geometric and mechanical characteristics having one to two overhangs with variable lengths. Murphy studied, by an algebraic solution approach, a simply supported beam with two overhangs of arbitrary length, allowing for an experimental determination of the elastic modulus E. The advantage of our article is that it offers the possibility of extending this approach to many interesting problems formed by transversely vibrating beams with various end constraints.Keywords: beam, finite element, transverse vibrations, end restreint, Bernoulli-Euler theory
Procedia PDF Downloads 7718642 Structural Element Vibration Analysis with finite element method: Use of Rayleigh Quotient
Authors: Houari Boumediene University of Science, Technology.
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"Various methods are typically used in the dynamic analysis of transversely vibrating beams. To achieve this, numerical methods are used to solve the general eigenvalue problem. The equations of equilibrium, which describe the motion, are derived from a fourth-order differential equation. Our study is based on the finite element method, and the results of the investigation are the vibration frequencies obtained using the Jacobi method. Two types of elementary mass matrices are considered: one representing a uniform distribution of mass along the element and the other consisting of concentrated masses located at fixed points whose number increases progressively with equal distances at each evaluation stage. The beams studied have different boundary constraints, representing several classical situations. Comparisons are made for beams where the distributed mass is replaced by n concentrated masses. As expected, the first calculation stage involves determining the lowest number of beam parts that gives a frequency comparable to that obtained from the Rayleigh formula. The obtained values are then compared to theoretical results based on the assumptions of the Bernoulli-Euler theory. These steps are repeated for the second type of mass representation in the same manner."Keywords: finite element method, bernouilli eulertheory, structural analysis, vibration analysis, rayleigh quotient
Procedia PDF Downloads 8118641 Several Spectrally Non-Arbitrary Ray Patterns of Order 4
Authors: Ling Zhang, Feng Liu
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A matrix is called a ray pattern matrix if its entries are either 0 or a ray in complex plane which originates from 0. A ray pattern A of order n is called spectrally arbitrary if the complex matrices in the ray pattern class of A give rise to all possible nth degree complex polynomial. Otherwise, it is said to be spectrally non-arbitrary ray pattern. We call that a spectrally arbitrary ray pattern A of order n is minimally spectrally arbitrary if any nonzero entry of A is replaced, then A is not spectrally arbitrary. In this paper, we find that is not spectrally arbitrary when n equals to 4 for any θ which is greater than or equal to 0 and less than or equal to n. In this article, we give several ray patterns A(θ) of order n that are not spectrally arbitrary for some θ which is greater than or equal to 0 and less than or equal to n. by using the nilpotent-Jacobi method. One example is given in our paper.Keywords: spectrally arbitrary, nilpotent matrix , ray patterns, sign patterns
Procedia PDF Downloads 16918640 Non-Local Behavior of a Mixed-Mode Crack in a Functionally Graded Piezoelectric Medium
Authors: Nidhal Jamia, Sami El-Borgi
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In this paper, the problem of a mixed-Mode crack embedded in an infinite medium made of a functionally graded piezoelectric material (FGPM) with crack surfaces subjected to electro-mechanical loadings is investigated. Eringen’s non-local theory of elasticity is adopted to formulate the governing electro-elastic equations. The properties of the piezoelectric material are assumed to vary exponentially along a perpendicular plane to the crack. Using Fourier transform, three integral equations are obtained in which the unknown variables are the jumps of mechanical displacements and electric potentials across the crack surfaces. To solve the integral equations, the unknowns are directly expanded as a series of Jacobi polynomials, and the resulting equations solved using the Schmidt method. In contrast to the classical solutions based on the local theory, it is found that no mechanical stress and electric displacement singularities are present at the crack tips when nonlocal theory is employed to investigate the problem. A direct benefit is the ability to use the calculated maximum stress as a fracture criterion. The primary objective of this study is to investigate the effects of crack length, material gradient parameter describing FGPMs, and lattice parameter on the mechanical stress and electric displacement field near crack tips.Keywords: functionally graded piezoelectric material (FGPM), mixed-mode crack, non-local theory, Schmidt method
Procedia PDF Downloads 30218639 Portfolio Selection with Constraints on Trading Frequency
Authors: Min Dai, Hong Liu, Shuaijie Qian
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We study a portfolio selection problem of an investor who faces constraints on rebalancing frequency, which is common in pension fund investment. We formulate it as a multiple optimal stopping problem and utilize the dynamic programming principle. By numerically solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation, we find a series of free boundaries characterizing optimal strategy, and the constraints significantly impact the optimal strategy. Even in the absence of transaction costs, there is a no-trading region, depending on the number of the remaining trading chances. We also find that the equivalent wealth loss caused by the constraints is large. In conclusion, our model clarifies the impact of the constraints on transaction frequency on the optimal strategy.Keywords: portfolio selection, rebalancing frequency, optimal strategy, free boundary, optimal stopping
Procedia PDF Downloads 7118638 A Fundamental Functional Equation for Lie Algebras
Authors: Ih-Ching Hsu
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Inspired by the so called Jacobi Identity (x y) z + (y z) x + (z x) y = 0, the following class of functional equations EQ I: F [F (x, y), z] + F [F (y, z), x] + F [F (z, x), y] = 0 is proposed, researched and generalized. Research methodologies begin with classical methods for functional equations, then evolve into discovering of any implicit algebraic structures. One of this paper’s major findings is that EQ I, under two additional conditions F (x, x) = 0 and F (x, y) + F (y, x) = 0, proves to be a fundamental functional equation for Lie Algebras. Existence of non-trivial solutions for EQ I can be proven by defining F (p, q) = [p q] = pq –qp, where p and q are quaternions, and pq is the quaternion product of p and q. EQ I can be generalized to the following class of functional equations EQ II: F [G (x, y), z] + F [G (y, z), x] + F [G (z, x), y] = 0. Concluding Statement: With a major finding proven, and non-trivial solutions derived, this research paper illustrates and provides a new functional equation scheme for studies in two major areas: (1) What underlying algebraic structures can be defined and/or derived from EQ I or EQ II? (2) What conditions can be imposed so that conditional general solutions to EQ I and EQ II can be found, investigated and applied?Keywords: fundamental functional equation, generalized functional equations, Lie algebras, quaternions
Procedia PDF Downloads 21718637 Machine Learning Analysis of Eating Disorders Risk, Physical Activity and Psychological Factors in Adolescents: A Community Sample Study
Authors: Marc Toutain, Pascale Leconte, Antoine Gauthier
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Introduction: Eating Disorders (ED), such as anorexia, bulimia, and binge eating, are psychiatric illnesses that mostly affect young people. The main symptoms concern eating (restriction, excessive food intake) and weight control behaviors (laxatives, vomiting). Psychological comorbidities (depression, executive function disorders, etc.) and problematic behaviors toward physical activity (PA) are commonly associated with ED. Acquaintances on ED risk factors are still lacking, and more community sample studies are needed to improve prevention and early detection. To our knowledge, studies are needed to specifically investigate the link between ED risk level, PA, and psychological risk factors in a community sample of adolescents. The aim of this study is to assess the relation between ED risk level, exercise (type, frequency, and motivations for engaging in exercise), and psychological factors based on the Jacobi risk factors model. We suppose that a high risk of ED will be associated with the practice of high caloric cost PA, motivations oriented to weight and shape control, and psychological disturbances. Method: An online survey destined for students has been sent to several middle schools and colleges in northwest France. This survey combined several questionnaires, the Eating Attitude Test-26 assessing ED risk; the Exercise Motivation Inventory–2 assessing motivations toward PA; the Hospital Anxiety and Depression Scale assessing anxiety and depression, the Contour Drawing Rating Scale; and the Body Esteem Scale assessing body dissatisfaction, Rosenberg Self-esteem Scale assessing self-esteem, the Exercise Dependence Scale-Revised assessing PA dependence, the Multidimensional Assessment of Interoceptive Awareness assessing interoceptive awareness and the Frost Multidimensional Perfectionism Scale assessing perfectionism. Machine learning analysis will be performed in order to constitute groups with a tree-based model clustering method, extract risk profile(s) with a bootstrap method comparison, and predict ED risk with a prediction method based on a decision tree-based model. Expected results: 1044 complete records have already been collected, and the survey will be closed at the end of May 2022. Records will be analyzed with a clustering method and a bootstrap method in order to reveal risk profile(s). Furthermore, a predictive tree decision method will be done to extract an accurate predictive model of ED risk. This analysis will confirm typical main risk factors and will give more data on presumed strong risk factors such as exercise motivations and interoceptive deficit. Furthermore, it will enlighten particular risk profiles with a strong level of proof and greatly contribute to improving the early detection of ED and contribute to a better understanding of ED risk factors.Keywords: eating disorders, risk factors, physical activity, machine learning
Procedia PDF Downloads 7318636 Socratic Style of Teaching: An Analysis of Dialectical Method
Authors: Muhammad Jawwad, Riffat Iqbal
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The Socratic method, also known as the dialectical method and elenctic method, has significant relevance in the contemporary educational system. It can be incorporated into modern-day educational systems theoretically as well as practically. Being interactive and dialogue-based in nature, this teaching approach is followed by critical thinking and innovation. The pragmatic value of the Dialectical Method has been discussed in this article, and the limitations of the Socratic method have also been highlighted. The interactive Method of Socrates can be used in many subjects for students of different grades. The Limitations and delimitations of the Method have also been discussed for its proper implementation. This article has attempted to elaborate and analyze the teaching method of Socrates with all its pre-suppositions and Epistemological character.Keywords: Socratic method, dialectical method, knowledge, teaching, virtue
Procedia PDF Downloads 12418635 Rogue Waves Arising on the Standing Periodic Wave in the High-Order Ablowitz-Ladik Equation
Authors: Yanpei Zhen
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The nonlinear Schrödinger (NLS) equation models wave dynamics in many physical problems related to fluids, plasmas, and optics. The standing periodic waves are known to be modulationally unstable, and rogue waves (localized perturbations in space and time) have been observed on their backgrounds in numerical experiments. The exact solutions for rogue waves arising on the periodic standing waves have been obtained analytically. It is natural to ask if the rogue waves persist on the standing periodic waves in the integrable discretizations of the integrable NLS equation. We study the standing periodic waves in the semidiscrete integrable system modeled by the high-order Ablowitz-Ladik (AL) equation. The standing periodic wave of the high-order AL equation is expressed by the Jacobi cnoidal elliptic function. The exact solutions are obtained by using the separation of variables and one-fold Darboux transformation. Since the cnoidal wave is modulationally unstable, the rogue waves are generated on the periodic background.Keywords: Darboux transformation, periodic wave, Rogue wave, separating the variables
Procedia PDF Downloads 17418634 Optimal Investment and Consumption Decision for an Investor with Ornstein-Uhlenbeck Stochastic Interest Rate Model through Utility Maximization
Authors: Silas A. Ihedioha
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In this work; it is considered that an investor’s portfolio is comprised of two assets; a risky stock which price process is driven by the geometric Brownian motion and a risk-free asset with Ornstein-Uhlenbeck Stochastic interest rate of return, where consumption, taxes, transaction costs and dividends are involved. This paper aimed at the optimization of the investor’s expected utility of consumption and terminal return on his investment at the terminal time having power utility preference. Using dynamic optimization procedure of maximum principle, a second order nonlinear partial differential equation (PDE) (the Hamilton-Jacobi-Bellman equation HJB) was obtained from which an ordinary differential equation (ODE) obtained via elimination of variables. The solution to the ODE gave the closed form solution of the investor’s problem. It was found the optimal investment in the risky asset is horizon dependent and a ratio of the total amount available for investment and the relative risk aversion coefficient.Keywords: optimal, investment, Ornstein-Uhlenbeck, utility maximization, stochastic interest rate, maximum principle
Procedia PDF Downloads 21518633 A Stochastic Volatility Model for Optimal Market-Making
Authors: Zubier Arfan, Paul Johnson
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The electronification of financial markets and the rise of algorithmic trading has sparked a lot of interest from the mathematical community, for the market making-problem in particular. The research presented in this short paper solves the classic stochastic control problem in order to derive the strategy for a market-maker. It also shows how to calibrate and simulate the strategy with real limit order book data for back-testing. The ambiguity of limit-order priority in back-testing is dealt with by considering optimistic and pessimistic priority scenarios. The model, although it does outperform a naive strategy, assumes constant volatility, therefore, is not best suited to the LOB data. The Heston model is introduced to describe the price and variance process of the asset. The Trader's constant absolute risk aversion utility function is optimised by numerically solving a 3-dimensional Hamilton-Jacobi-Bellman partial differential equation to find the optimal limit order quotes. The results show that the stochastic volatility market-making model is more suitable for a risk-averse trader and is also less sensitive to calibration error than the constant volatility model.Keywords: market-making, market-microsctrucure, stochastic volatility, quantitative trading
Procedia PDF Downloads 13818632 A New Computational Package for Using in CFD and Other Problems (Third Edition)
Authors: Mohammad Reza Akhavan Khaleghi
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This paper shows changes done to the Reduced Finite Element Method (RFEM) that its result will be the most powerful numerical method that has been proposed so far (some forms of this method are so powerful that they can approximate the most complex equations simply Laplace equation!). Finite Element Method (FEM) is a powerful numerical method that has been used successfully for the solution of the existing problems in various scientific and engineering fields such as its application in CFD. Many algorithms have been expressed based on FEM, but none have been used in popular CFD software. In this section, full monopoly is according to Finite Volume Method (FVM) due to better efficiency and adaptability with the physics of problems in comparison with FEM. It doesn't seem that FEM could compete with FVM unless it was fundamentally changed. This paper shows those changes and its result will be a powerful method that has much better performance in all subjects in comparison with FVM and another computational method. This method is not to compete with the finite volume method but to replace it.Keywords: reduced finite element method, new computational package, new finite element formulation, new higher-order form, new isogeometric analysis
Procedia PDF Downloads 10518631 A Study on the Solutions of the 2-Dimensional and Forth-Order Partial Differential Equations
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In this study, we will carry out a comparative study between the reduced differential transform method, the adomian decomposition method, the variational iteration method and the homotopy analysis method. These methods are used in many fields of engineering. This is been achieved by handling a kind of 2-Dimensional and forth-order partial differential equations called the Kuramoto–Sivashinsky equations. Three numerical examples have also been carried out to validate and demonstrate efficiency of the four methods. Furthermost, it is shown that the reduced differential transform method has advantage over other methods. This method is very effective and simple and could be applied for nonlinear problems which used in engineering.Keywords: reduced differential transform method, adomian decomposition method, variational iteration method, homotopy analysis method
Procedia PDF Downloads 42018630 Elvis Improved Method for Solving Simultaneous Equations in Two Variables with Some Applications
Authors: Elvis Adam Alhassan, Kaiyu Tian, Akos Konadu, Ernest Zamanah, Michael Jackson Adjabui, Ibrahim Justice Musah, Esther Agyeiwaa Owusu, Emmanuel K. A. Agyeman
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In this paper, how to solve simultaneous equations using the Elvis improved method is shown. The Elvis improved method says; to make one variable in the first equation the subject; make the same variable in the second equation the subject; equate the results and simplify to obtain the value of the unknown variable; put the value of the variable found into one equation from the first or second steps and simplify for the remaining unknown variable. The difference between our Elvis improved method and the substitution method is that: with Elvis improved method, the same variable is made the subject in both equations, and the two resulting equations equated, unlike the substitution method where one variable is made the subject of only one equation and substituted into the other equation. After describing the Elvis improved method, findings from 100 secondary students and the views of 5 secondary tutors to demonstrate the effectiveness of the method are presented. The study's purpose is proved by hypothetical examples.Keywords: simultaneous equations, substitution method, elimination method, graphical method, Elvis improved method
Procedia PDF Downloads 11918629 Different Methods of Fe3O4 Nano Particles Synthesis
Authors: Arezoo Hakimi, Afshin Farahbakhsh
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Herein, we comparison synthesized Fe3O4 using, hydrothermal method, Mechanochemical processes and solvent thermal method. The Hydrothermal Technique has been the most popular one, gathering interest from scientists and technologists of different disciplines, particularly in the last fifteen years. In the hydrothermal method Fe3O4 microspheres, in which many nearly monodisperse spherical particles with diameters of about 400nm, in the mechanochemical method regular morphology indicates that the particles are well crystallized and in the solvent thermal method Fe3O4 nanoparticles have good properties of uniform size and good dispersion.Keywords: Fe3O4 nanoparticles, hydrothermal method, mechanochemical processes, solvent thermal method
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