Search results for: Boolean equations

Authors: A. M. Sagir

Abstract:

Discrete linear multistep block method of uniform order for the solution of first order Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) is presented in this paper. The approach of interpolation and collocation approximation are adopted in the derivation of the method which is then applied to first order ordinary differential equations with associated initial conditions. The continuous hybrid formulations enable us to differentiate and evaluate at some grids and off – grid points to obtain four discrete schemes, which were used in block form for parallel or sequential solutions of the problems. Furthermore, a stability analysis and efficiency of the block method are tested on ordinary differential equations, and the results obtained compared favorably with the exact solution.

Keywords: block method, first order ordinary differential equations, hybrid, self-starting

Procedia PDF Downloads 482

Authors: Jyh-Yang Wu, Sheng-Gwo Chen

Abstract:

In this paper, we present a simple effective numerical geometric method to estimate the divergence of a vector field over a curved surface. The conservation law is an important principle in physics and mathematics. However, many well-known numerical methods for solving diffusion equations do not obey conservation laws. Our presented method in this paper combines the divergence theorem with a generalized finite difference method and obeys the conservation law on discrete closed surfaces. We use the similar method to solve the Cahn-Hilliard equations on evolving spherical surfaces and observe stability results in our numerical simulations.

Keywords: conservation laws, diffusion equations, Cahn-Hilliard equations, evolving surfaces

Procedia PDF Downloads 494

Authors: F. Maass, P. Martin, J. Olivares

Abstract:

The aim of this work is the application of the program GeoGebra in teaching the study of nonhomogeneous linear second order differential equations with constant coefficients. Different kind of functions or forces will be considered in the right hand side of the differential equations, in particular, the emphasis will be placed in the case of trigonometrical functions producing the resonance phenomena. In order to obtain this, the frequencies of the trigonometrical functions will be changed. Once the resonances appear, these have to be correlationated with the roots of the second order algebraic equation determined by the coefficients of the differential equation. In this way, the physics and engineering students will understand resonance effects and its consequences in the simplest way. A large variety of examples will be shown, using different kind of functions for the nonhomogeneous part of the differential equations.

Keywords: education, geogebra, ordinary differential equations, resonance

Procedia PDF Downloads 245

Authors: Armin Ardekani, Mohammad Akbari

Abstract:

We present a method of generating series solutions to large classes of nonlinear differential equations. The method is well suited to be adapted in mathematical software and unlike the available commercial solvers, we are capable of generating solutions to boundary value ODEs and PDEs. Many of the generated solutions converge to closed form solutions. Our method can also be applied to systems of ODEs or PDEs, providing all the solutions efficiently. As examples, we present results to many difficult differential equations in engineering fields.

Keywords: computational mathematics, differential equations, engineering, series

Procedia PDF Downloads 336

Authors: Kholod Mohammad Abualnaja

Abstract:

A new algorithm is presented to find some new iterative methods for solving nonlinear equations F(x)=0 by using the variational iteration method. The efficiency of the considered method is illustrated by example. The results show that the proposed iteration technique, without linearization or small perturbation, is very effective and convenient.

Keywords: variational iteration method, nonlinear equations, Lagrange multiplier, algorithms

Procedia PDF Downloads 545

Authors: Rabia Khan, Nargis Munir, Suriya Gharib, Syeda Roshana Ali

Abstract:

In this paper linear equations are discussed in detail along with elimination method. Gaussian elimination and Gauss Jordan schemes are carried out to solve the linear system of equation. This paper comprises of matrix introduction, and the direct methods for linear equations. The goal of this research was to analyze different elimination techniques of linear equations and measure the performance of Gaussian elimination and Gauss Jordan method, in order to find their relative importance and advantage in the field of symbolic and numeric computation. The purpose of this research is to revise an introductory concept of linear equations, matrix theory and forms of Gaussian elimination through which the performance of Gauss Jordan and Gaussian elimination can be measured.

Keywords: direct, indirect, backward stage, forward stage

Procedia PDF Downloads 596

Authors: Baghdad Said

Abstract:

Differential equations of fractional order have proved to be important tools to describe many physical phenomena and have been used in diverse fields such as engineering, mathematics as well as other applied sciences. On the other hand, the theory of differential equations involving the Stieltjes derivative (SD) with respect to a non-decreasing function is a new class of differential equations and has many applications as a unified framework for dynamic equations on time scales and differential equations with impulses at fixed times. The aim of this paper is to investigate the existence, uniqueness, and generalized Ulam-Hyers-Rassias stability (UHRS) of solutions for a boundary value problem of sequential fractional differential equations (SFDE) containing (SD). This study is based on the technique of noncompactness measures (MNCs) combined with Monch-Krasnoselski fixed point theorems (FPT), and the results are proven in an appropriate Banach space under sufficient hypotheses. We also give an illustrative example. In this work, we introduced a class of (SFDE) and the results are obtained under a few hypotheses. Future directions connected to this work could focus on another problem with different types of fractional integrals and derivatives, and the (SD) will be assumed under a more general hypothesis in more general functional spaces.

Keywords: SFDE, SD, UHRS, MNCs, FPT

Procedia PDF Downloads 41

Authors: Zhanar Imanova

Abstract:

Masses of real celestial bodies change anisotropically and reactive forces appear, and they need to be taken into account in the study of these bodies' dynamics. We studied the two-planet problem of three bodies with variable masses in the presence of reactive forces and obtained the equations of perturbed motion in Newton’s form equations. The motion equations in the orbital coordinate system, unlike the Lagrange equation, are convenient for taking into account the reactive forces. The perturbing force is expanded in terms of osculating elements. The expansion of perturbing functions is a time-consuming analytical calculation and results in very cumber some analytical expressions. In the considered problem, we obtained expansions of perturbing functions by small parameters up to and including the second degree. In the non resonant case, we obtained evolution equations in the Newton equation form. All symbolic calculations were done in Wolfram Mathematica.

Keywords: two-planet, three-body problem, variable mass, evolutionary equations

Procedia PDF Downloads 65

Authors: Matevž Breška, Iztok Peruš, Vlado Stankovski

Abstract:

Systematic overview of existing Ground Motion Prediction Equations (GMPEs) has been published by Douglas. The number of earthquake recordings that have been used for fitting these equations has increased in the past decades. The current PF-L database contains 3550 recordings. Since the GMPEs frequently model the peak ground acceleration (PGA) the goal of the present study was to refit a selection of 44 of the existing equation models for PGA in light of the latest data. The algorithm Levenberg-Marquardt was used for fitting the coefficients of the equations and the results are evaluated both quantitatively by presenting the root mean squared error (RMSE) and qualitatively by drawing graphs of the five best fitted equations. The RMSE was found to be as low as 0.08 for the best equation models. The newly estimated coefficients vary from the values published in the original works.

Keywords: Ground Motion Prediction Equations, Levenberg-Marquardt algorithm, refitting PF-L database, peak ground acceleration

Procedia PDF Downloads 462

Authors: Zahid Ullah, Atlas Khan

Abstract:

The presented research is dedicated to an extensive examination of the regularity properties associated with a specific class of equations, namely extremely degenerate elliptic equations. This study holds significance in unraveling the complexities inherent in these equations and understanding the smoothness of their solutions. The focus is on analyzing the regularity of results, aiming to contribute to the broader field of mathematical theory. By delving into the intricacies of extremely degenerate elliptic equations, the research seeks to advance our understanding beyond conventional analyses, addressing challenges posed by degeneracy and pushing the boundaries of classical analytical methods. The motivation for this exploration lies in the practical applicability of mathematical models, particularly in real-world scenarios where physical phenomena exhibit characteristics that challenge traditional mathematical modeling. The research aspires to fill gaps in the current understanding of regularity properties within solutions to extremely degenerate elliptic equations, ultimately contributing to both theoretical foundations and practical applications in diverse scientific fields.

Keywords: investigating smoothness, extremely degenerate elliptic equations, regularity properties, mathematical analysis, complexity solutions

Procedia PDF Downloads 60

Authors: A. M. Sagir

Abstract:

This paper derived four newly schemes which are combined in order to form an accurate and efficient block method for parallel or sequential solution of third order ordinary differential equations of the form y^'''= f(x,y,y^',y^'' ), y(α)=y_0,〖y〗^' (α)=β,y^('' ) (α)=μ with associated initial or boundary conditions. The implementation strategies of the derived method have shown that the block method is found to be consistent, zero stable and hence convergent. The derived schemes were tested on stiff and non-stiff ordinary differential equations, and the numerical results obtained compared favorably with the exact solution.

Keywords: block method, hybrid, linear multistep, self-starting, third order ordinary differential equations

Procedia PDF Downloads 271

Authors: Serdar Kucuk, Redwan Alqasemi

Abstract:

In this paper, the generic kinematic equations of 1-Degrees-Of-Freedom (DOF), 2-DOF and 3-DOF above-knee prosthetic legs are derived using the mathematical tools used in science of robotics. As it is known, since the human leg performs rotational motions in the knee joint and foot-ankle joint, the axial rotational motions in the above-knee prosthetic legs are performed by using one or more revolute joints. When deriving the kinematic equations of the 1-DOF, 2-DOF, and 3-DOF above-knee prosthetic legs, the foot-ankle is treated as if there were a fixed non-rotating joint, a revolute joint, and a universal joint, respectively. The kinematic equations of the prosthetic legs presented in this article are obtained using DH method. The main advantages of this method are the easy physical interpretation of robot mechanisms and the use of 4x4 homogeneous transformation matrices, which are widely used in the literature. It is thought that the equations presented in this article contribute positively to the design, control, simulation and hence easy production of above-knee prosthetic legs.

Keywords: robotic above-knee prosthetic legs, generic kinematic equations, revolute and universal joints, DH method

Procedia PDF Downloads 0

Authors: Melusi Khumalo, Anastacia Dlamini

Abstract:

In this paper, we consider a numerical solution for nonlinear Fredholm integral equations of the second kind. We work with uniform mesh and use the Lagrange polynomials together with the Galerkin finite element method, where the weight function is chosen in such a way that it takes the form of the approximate solution but with arbitrary coefficients. We implement the finite element method to the nonlinear Fredholm integral equations of the second kind. We consider the error analysis of the method. Furthermore, we look at a specific example to illustrate the implementation of the finite element method.

Keywords: finite element method, Galerkin approach, Fredholm integral equations, nonlinear integral equations

Procedia PDF Downloads 376

Authors: K. P. Mredula, D. C. Vakaskar

Abstract:

The article traces developments and evolution of various algorithms developed for solving partial differential equations using the significant combination of wavelet with few already explored solution procedures. The approach depicts a study over a decade of traces and remarks on the modifications in implementing multi-resolution of wavelet, finite difference approach, finite element method and finite volume in dealing with a variety of partial differential equations in the areas like plasma physics, astrophysics, shallow water models, modified Burger equations used in optical fibers, biology, fluid dynamics, chemical kinetics etc.

Keywords: multi-resolution, Haar Wavelet, partial differential equation, numerical methods

Procedia PDF Downloads 299

Authors: Archit Yajnik, Rustam Ali

Abstract:

In this paper, numerical method based on discrete GHM multiwavelets is presented for solving the Fredholm integral equations of second kind. There is hardly any article available in the literature in which the integral equations are numerically solved using discrete GHM multiwavelet. A number of examples are demonstrated to justify the applicability of the method. In GHM multiwavelets, the values of scaling and wavelet functions are calculated only at t = 0, 0.5 and 1. The numerical solution obtained by the present approach is compared with the traditional Quadrature method. It is observed that the present approach is more accurate and computationally efficient as compared to quadrature method.

Keywords: GHM multiwavelet, fredholm integral equations, quadrature method, function approximation

Procedia PDF Downloads 462

Authors: Minas Balyan

Abstract:

Nowadays X-ray nonlinear diffraction and nonlinear effects are investigated due to the presence of the third generation synchrotron sources and XFELs. X-ray third order nonlinear dynamical diffraction is considered as well. Using the nonlinear model of the usual visible light optics the third-order nonlinear Takagi’s equations for monochromatic waves and the third-order nonlinear time-dependent dynamical diffraction equations for X-ray pulses are obtained by the author in previous papers. The obtained equations show, that even if the Fourier-coefficients of the linear and the third order nonlinear susceptibilities are zero (forbidden reflection), the dynamical diffraction in the nonlinear case is related to the presence in the nonlinear equations the terms proportional to the zero order and the second order nonzero Fourier coefficients of the third order nonlinear susceptibility. Thus, in the third order nonlinear Bragg diffraction case a nonlinear analogue of the well-known Renninger effect takes place. In this work, the 'third order nonlinear Renninger effect' is considered theoretically.

Keywords: Bragg diffraction, nonlinear Takagi’s equations, nonlinear Renninger effect, third order nonlinearity

Procedia PDF Downloads 385

Authors: Farid Nouioua, Abdelouaheb Ardjouni

Abstract:

In this article, we study the existence of positive periodic solutions of certain delay differential equations. In the process we convert the differential equation into an equivalent integral equation after which appropriate mappings are constructed. We then employ Krasnoselskii's fixed point theorem to obtain sufficient conditions for the existence of a positive periodic solution of the differential equation. The obtained results improve and extend the results in the literature. Finally, an example is given to illustrate our results.

Keywords: delay differential equations, positive periodic solutions, integral equations, Krasnoselskii fixed point theorem

Procedia PDF Downloads 438

Authors: T. Bouchabchoub, A. Bendahmane, A. Haouriqui, N. Attou

Abstract:

This paper presents simulation results of Forex predicting model equations in order to give approximately a prevision of interest rates. First, Hall-Taylor (HT) equations have been used with Taylor rule (TR) to adapt them to European and American Forex Markets. Indeed, initial Taylor Rule equation is conceived for all Forex transactions in every States: It includes only one equation and six parameters. Here, the model has been used with Hall-Taylor equations, initially including twelve equations which have been reduced to only three equations. Analysis has been developed on the following base macroeconomic variables: Real change rate, investment wages, anticipated inflation, realized inflation, real production, interest rates, gap production and potential production. This model has been used to specifically study the impact of an inflation shock on macroeconomic director interest rates.

Keywords: interest rate, Forex, Taylor rule, production, European Central Bank (ECB), Federal Reserve System (FED).

Procedia PDF Downloads 527

Authors: Adetunji A. Adeyanju., Mathew O. Omeike, Johnson O. Adeniran, Biodun S. Badmus

Abstract:

In this paper, we discuss certain conditions for uniform asymptotic stability and uniform ultimate boundedness of solutions to some systems of Aizermann-type of differential equations by means of second method of Lyapunov. In achieving our goal, some Lyapunov functions are constructed to serve as basic tools. The stability results in this paper, extend some stability results for some Aizermann-type of differential equations found in literature. Also, we prove some results on uniform boundedness and uniform ultimate boundedness of solutions of systems of equations study.

Keywords: Aizermann, boundedness, first order, Lyapunov function, stability

Procedia PDF Downloads 84

Authors: Zuhier Altawallbeh

Abstract:

This paper investigates the approximate analytical solutions of general form of Volterra integro-differential equations system by using the residual power series method (for short RPSM). The proposed method produces the solutions in terms of convergent series requires no linearization or small perturbation and reproduces the exact solution when the solution is polynomial. Some examples are given to demonstrate the simplicity and efficiency of the proposed method. Comparisons with the Laplace decomposition algorithm verify that the new method is very effective and convenient for solving system of pantograph equations.

Keywords: integro-differential equation, pantograph equations, system of initial value problems, residual power series method

Procedia PDF Downloads 418

Authors: Mustafa Ekici

Abstract:

Natural convection is a basic process which is important in a wide variety of practical applications. In essence, a heated fluid expands and rises from buoyancy due to decreased density. Numerous papers have been written on natural or mixed convection in vertical ducts heated on the side. These equations have been proved to be valuable tools for the modelling of many phenomena such as fluid dynamics. Finding solutions to such equations or system of equations are in general not an easy task. We propose a method, which is called differential transform method, of solving a non-linear equations and compare the results with some of the other techniques. Illustrative examples shows that the results are in good agreement.

Keywords: differential transform method, momentum, energy equation, boundry value problem

Procedia PDF Downloads 461

Authors: Meziane Belkacem

Abstract:

We aim at converting the original 3D Lorenz-Haken equations, which describe laser dynamics –in terms of self-pulsing and chaos- into 2-second-order differential equations, out of which we extract the so far missing mathematics and corroborations with respect to nonlinear interactions. Leaning on basic trigonometry, we pull out important outcomes; a fundamental result attributes chaos to forbidden periodic solutions inside some precisely delimited region of the control parameter space that governs the bewildering dynamics.

Keywords: Physics, optics, nonlinear dynamics, chaos

Procedia PDF Downloads 157

Authors: Andriy Didenko, Zanin Kavazovic

Abstract:

Engineering students often have certain difficulties in mastering major theoretical concepts in mathematical courses such as differential equations. Student projects were proposed to motivate students’ learning and can be used as a tool to promote students’ interest in the material. Authors propose a student project that includes the use of Microsoft Excel. This instructional tool is often overlooked by both educators and students. An integral component of the experimental part of such a project is the exploration of an interactive spreadsheet. The aim is to assist engineering students in better understanding of Euler’s method. This method is employed to numerically solve first order differential equations. At first, students are invited to select classic equations from a list presented in a form of a drop-down menu. For each of these equations, students can select and modify certain key parameters and observe the influence of initial condition on the solution. This will give students an insight into the behavior of the method in different configurations as solutions to equations are given in numerical and graphical forms. Further, students could also create their own equations by providing functions of their own choice and a variety of initial conditions. Moreover, they can visualize and explore the impact of the length of the time step on the convergence of a sequence of numerical solutions to the exact solution of the equation. As a final stage of the project, students are encouraged to develop their own spreadsheets for other numerical methods and other types of equations. Such projects promote students’ interest in mathematical applications and further improve their mathematical and programming skills.

Keywords: student project, Euler's method, spreadsheet, engineering education

Procedia PDF Downloads 135

Authors: Gaspar J. Machado, Stéphane Clain, Raphael Loubère

Abstract:

The goal of the present work is to numerically study steady-state nonlinear hyperbolic equations in the context of the finite volume framework. We will consider the unidimensional Burgers' equation as the reference case for the scalar situation and the unidimensional Euler equations for the vectorial situation. We consider two approaches to solve the nonlinear equations: a time marching algorithm and a direct steady-state approach. We first develop the necessary and sufficient conditions to obtain the existence and unicity of the solution. We treat regular examples and solutions with a steady shock and to provide very-high-order finite volume approximations we implement a method based on the MOOD technology (Multi-dimensional Optimal Order Detection). The main ingredient consists in using an 'a posteriori' limiting strategy to eliminate non physical oscillations deriving from the Gibbs phenomenon while keeping a high accuracy for the smooth part.

Keywords: Euler equations, finite volume, MOOD, steady-state

Procedia PDF Downloads 277

Authors: Fakhreddin Abedi, Wah June Leong

Abstract:

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

Procedia PDF Downloads 52

Authors: Zahid Ullah, Atlas Khan

Abstract:

This research endeavors to explore the regularity properties associated with a specific class of equations, namely extremely degenerate elliptic equations. These equations hold significance in understanding complex physical systems like porous media flow, with applications spanning various branches of mathematics. The focus is on unraveling and analyzing regularity results to gain insights into the smoothness of solutions for these highly degenerate equations. Elliptic equations, fundamental in expressing and understanding diverse physical phenomena through partial differential equations (PDEs), are particularly adept at modeling steady-state and equilibrium behaviors. However, within the realm of elliptic equations, the subset of extremely degenerate cases presents a level of complexity that challenges traditional analytical methods, necessitating a deeper exploration of mathematical theory. While elliptic equations are celebrated for their versatility in capturing smooth and continuous behaviors across different disciplines, the introduction of degeneracy adds a layer of intricacy. Extremely degenerate elliptic equations are characterized by coefficients approaching singular behavior, posing non-trivial challenges in establishing classical solutions. Still, the exploration of extremely degenerate cases remains uncharted territory, requiring a profound understanding of mathematical structures and their implications. The motivation behind this research lies in addressing gaps in the current understanding of regularity properties within solutions to extremely degenerate elliptic equations. The study of extreme degeneracy is prompted by its prevalence in real-world applications, where physical phenomena often exhibit characteristics defying conventional mathematical modeling. Whether examining porous media flow or highly anisotropic materials, comprehending the regularity of solutions becomes crucial. Through this research, the aim is to contribute not only to the theoretical foundations of mathematics but also to the practical applicability of mathematical models in diverse scientific fields.

Keywords: elliptic equations, extremely degenerate, regularity results, partial differential equations, mathematical modeling, porous media flow

Procedia PDF Downloads 73

Authors: Davood Yazdani Cherati, Hussein Hashemi Senejani

Abstract:

In this paper, a convenient analytical solution for a system of coupled differential equations, derived from thermo-hydro-mechanical analysis of three-phase porous media such as unsaturated soils is developed. This kind of analysis can be used in various fields such as geothermal energy systems and seepage of leachate from buried municipal and domestic waste in geomaterials. Initially, a system of coupled differential equations, including energy, mass, and momentum conservation equations is considered, and an analytical method called AGM is employed to solve the problem. The method is straightforward and comprehensible and can be used to solve various nonlinear partial differential equations (PDEs). Results indicate the accuracy of the applied method for solving nonlinear partial differential equations.

Keywords: AGM, analytical solution, porous media, thermo-hydro-mechanical, unsaturated soils

Procedia PDF Downloads 229

Authors: Jessada Tariboon

Abstract:

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

Procedia PDF Downloads 489

Authors: Trilok Mathur, Shivi Agarwal

Abstract:

This paper deals with the solutions of fuzzy-fractional-order Riccati equations under Caputo-type fuzzy fractional derivatives. The Caputo-type fuzzy fractional derivatives are defined based on Hukuhura difference and strongly generalized fuzzy differentiability. The Laplace-Adomian-Pade method is used for solving fractional Riccati-type initial value differential equations of fractional order. Moreover, we also displayed some examples to illustrate our methods.

Keywords: Caputo-type fuzzy fractional derivative, Fractional Riccati differential equations, Laplace-Adomian-Pade method, Mittag Leffler function

Procedia PDF Downloads 395

Authors: Y. N. Reddy

Abstract:

The class of differential-difference equations which have characteristics of both classes, i.e., delay/advance and singularly perturbed behaviour is known as singularly perturbed differential-difference equations. The expression ‘positive shift’ and ‘negative shift’ are also used for ‘advance’ and ‘delay’ respectively. In general, an ordinary differential equation in which the highest order derivative is multiplied by a small positive parameter and containing at least one delay/advance is known as singularly perturbed differential-difference equation. Singularly perturbed differential-difference equations arise in the modelling of various practical phenomena in bioscience, engineering, control theory, specifically in variational problems, in describing the human pupil-light reflex, in a variety of models for physiological processes or diseases and first exit time problems in the modelling of the determination of expected time for the generation of action potential in nerve cells by random synaptic inputs in dendrites. In this paper, we envisage the use of Liouville Green Transformation to find the solution of singularly perturbed differential difference equations. First, using Taylor series, the given singularly perturbed differential difference equation is approximated by an asymptotically equivalent singularly perturbation problem. Then the Liouville Green Transformation is applied to get the solution. Several model examples are solved, and the results are compared with other methods. It is observed that the present method gives better approximate solutions.

Keywords: difference equations, differential equations, singular perturbations, boundary layer

Procedia PDF Downloads 199