Application of Laplace Transform in Science and Engineering

One reliable mathematical tool that is used extensively in many scientific and technical fields is the Laplace transform (LP). Similar to the application of transfer functions in solving ordinary differential equations (ODEs), LPs offer a simple method for tackling increasingly complex engineering problems. LPs are used in physics and engineering, and this research first looks at such uses before concentrating on how they are used in electric circuit analysis. The research also explores more sophisticated uses, such as load frequency control in power systems engineering

Explicit Solution of First-Order Differential Equation Using Aitken’s and Newton’s Interpolation Methods

The struggle to find the analytic solution of several differential equations leads to several issues like difficulty in finding solutions, singularities, convergent issues, and stability. Because of these problems, most of the researchers come up with explicit approaches such as the Runge Kutta method, Euler’s method, and Taylor’s polynomial method for finding numerical solutions to the ordinary differential equation. In this work, we combine both the Aitken methods and Newton’s interpolation method (NIM) to solve first-order differential equations. The numerical results obtained provide minimal error. The result is supported by solving an example

Application of Lagrange Interpolation Method to Solve First-Order Differential Equation Using Newton Interpolation Approach

One of the important problems in mathematics is finding the analytic solution and numerical solution of the differential equation using various methods and techniques. Most of the researchers tackled different numerical approaches to solve ordinary differential equations. These methods such as the Runge Kutta method, Euler’s method, and Taylor’s polynomial method have so many issues like difficulties in finding the solution that can lead to singularities or no solution. In this work, we considered Newton’s interpolation and Lagrange’s interpolation polynomial method (LIPM). These studies combine both Newton’s interpolation method and Lagrange method (NIPM) to solve first-order differential equations. The results obtained provide minimum approximative error. The result is supported by solving an example

Solution of First-Order Differential Equation Using Fourth-Order Runge-Kutta Approachand Adams Bashforth Methods

In this research, we investigate the solution of first-order differential equations(DEs) using Runge- Kutta fourth-order method (RKM) and Adams-Bashforth methods (ABMs). In this work we considerfourth-order RKM and ABMs for solving first order DEs. The method proof to be simple, easy, accurate and efficient technique for solving first order DEs. Moreover, there are unlimited application of fourth-order RK4 and ABMs for solving first-order DE in science, engineering, economics, social science, biology and business. These play an important role in science and engineering. Some examples are giving and solved to support the efficiency of our methods which are demonstrated by figures

Solution of First-Order Differential Equation Using Fourth-Order Runge-Kutta Approach and Adams Bashforth Methods

In this research, we investigate the solution of first-order differential equations (DEs) using Runge- Kutta fourth-order method (RKM) and Adams-Bashforth methods (ABMs). In this work we consider fourth-order RKM and ABMs for solving first order DEs. The method proof to be simple, easy, accurate and efficient technique for solving first order DEs. Moreover, there are unlimited application of fourth-order RK4 and ABMs for solving first-order DE in science, engineering, economics, social science, biology and business. These play an important role in science and engineering. Some examples are giving and solved to support the efficiency of our methods which are demonstrated by figures. &nbsp

Numerical Approximation Method for Solving Differential Equations

This paper investigates numerical methods for solving differential equation. In this work, the continuous least square method (CLSM) was considered to find the best numerical approximation by solving differential equations. The continuous least square method (CLSM) was developed together with the L_2 norm. Numerical results obtained yield minimum approximation error, provide the best approximation. Explicit results obtained are supported by examples treated with MATLAB and Wolfram Mathematica 11

Solution for Second-Order Differential Equation Using Least Square Method

This paper studies the numerical method for solving differential equations. The continuous least square method (CLSM) is used to obtain the explicit solution for solving ordinary differential equations (ODEs), partial differential equations (PDEs), and fractional differential equations (FDEs), but in this work, we consider the explicit results from CLSM approach and applied it on second-order ODEs. Moreover, the L_2 norm is used to obtain the minimum approximation error. The numerical results obtained has a good agreement with the exact solution with minimum approximation error. The explicit results are supported by an example that was treated with Matlab and Matematica 11

Procesado de geometría en CAGD mediante S-series

El diseño geométrico asistido por ordenador (CAGD) se basa en la representación de entidades geométricas en el estándar nurbs, por lo que se debe obtener una aproximación polinómica o racional de aquellas funciones trascendentes, entidades que no pueden ser expresadas en la base de Bernstein. En principio se podría pensar en una aproximación mediante series de Taylor truncadas. De esta forma se obtendría una buena aproximación alrededor de un punto, pero se precisarían grados muy elevados para errores pequeños y los programas de CAD tienen limitado el grado maximo admisible. Una forma de evitar estos grados elevados seria conectar varios desarrollos de Taylor, pero en este caso aparecerían huecos en la unión de dos expansiones, algo inaceptable en una representación para CAD. En esta tesis se introduce la herramienta matemática básica empleada en este trabajo, las s-series. Estas series resultan de la base s-monomial, basada en expansiones de hermite en un intervalo unitario de la variable. Asimismo, se describen las estrategias para calcular de manera eficiente la aproximación de una entidad mediante s-series. Seguidamente, se comparan las aproximaciones mediante s-series con las basadas en series de poisson. A continuación, se aproxima la clotoide como ejemplo de aplicación de las estrategias de aproximación mediante s-series expuestas. Finalmente, se aplican las s-series a las técnicas de deformación. El objetivo de este capítulo consiste en conseguir una aproximación polinómica Bernstein-Bezier de los objetos deformados

تمثيل الإطار الخارجي للكلمات العربية بكفاءة من خلال الدمج بين نموذج الكنتور النشط وتحديد ونقاط الزوايا

Graphical curves and surfaces fitting are hot areas of research studies and application, such as artistic applications, analysis applications and encoding purposes. Outline capture of digital word images is important in most of the desktop publishing systems. The shapes of the characters are stored in the computer memory in terms of their outlines, and the outlines are expressed as Bezier curves. Existing methods for Arabic font outline description suffer from low fitting accuracy and efficiency. In our research, we developed a new method for outlining shapes using Bezier curves with minimal set of curve points. A distinguishing characteristic of our method is that it combines the active contour method (snake) with corner detection to achieve an initial set of points that is as close to the shape's boundaries as possible. The method links these points (snake + corner) into a compound Bezier curve, and iteratively improves the fitting of the curve over the actual boundaries of the shape. We implemented and tested our method using MATLAB. Test cases included various levels of shape complexity varying from simple, moderate, and high complexity depending on factors, such as: boundary concavities, number of corners. Results show that our method achieved average 86% of accuracy when measured relative to true shape boundary. When compared to other similar methods (Masood & Sarfraz, 2009; Sarfraz & Khan, 2002; Ferdous A Sohel, Karmakar, Dooley, & Bennamoun, 2010), our method performed comparatively well. Keywords: Bezier curves, shape descriptor, curvature, corner points, control points, Active Contour Model.تعتبر المنحنيات والأسطح الرسومية موضوعاً هاماً في الدراسات البحثية وفي التطبيقات البرمجية مثل التطبيقات الفنية، وتطبيقات تحليل وترميز البيانات. ويعتبر تخطيط الحدود الخارجية للكلمات عملية أساسية في غالبية تطبيقات النشر المكتبي. في هذه التطبيقات تخزن أشكال الأحرف في الذاكرة من حيث خطوطها الخارجية، وتمثل الخطوط الخارجية على هيئة منحنيات Bezier. الطرق المستخدمة حالياً لتحديد الخطوط الخارجية للكلمات العربية تنقصها دقة وكفاءة الملاءمة ما بين الحدود الحقيقية والمنحنى الرسومي الذي تقوم بتشكيله. في هذا البحث قمنا بتطوير طريقة جديدة لتخطيط الحدود الخارجية للكلمات تعتمد على منحنيات Bezier بمجموعة أقل من المنحنيات الجزئية. تتميز طريقتنا بخاصية مميزة وهي الدمج بين آلية لاستشعار الزوايا مع آلية نموذج الكنتور النشط (الأفعى). يتم الدمج بين نقاط الزوايا ونقاط الأفعى لتشكيل مجموعة موحدة من النقاط المبدئية قريبة قدر الإمكان من الحدود الحقيقية للشكل المراد تحديده. يتشكل منحنى Bezier من هذه المجموعة المدمجة، وتتم عملية تدريجية على دورات لملاءمة المنحنى على الحدود الحقيقية للشكل. قام الباحث بتنفيذ وتجربة الطريقة الجديدة باستخدام برنامج MATLAB. وتم اختيار أشكال رسومية كعينات اختبار تتصف بمستويات متباينة من التعقيد تتراوح ما بين بسيط إلى متوسط إلى عالي التعقيد على أساس عوامل مثل تقعرات الحدود، عدد نقاط الزوايا، الفتحات الداخلية، إلخ. وقد أظهرت نتائج الاختبار أن طريقتنا الجديدة حققت دقة في الملائمة تصل نسبتها إلى 86% مقارنة بالحدود الحقيقية للشكل المستهدف. وكذلك فقد كان أداء طريقتنا جيداً بالمقارنة مع طرق أخرى مماثلة