Backstepping PDE Design: A Convex Optimization Approach

Abstract\u2014Backstepping design for boundary linear PDE is formulated as a convex optimization problem. Some classes of parabolic PDEs and a first-order hyperbolic PDE are studied, with particular attention to non-strict feedback structures. Based on the compactness of the Volterra and Fredholm-type operators involved, their Kernels are approximated via polynomial functions. The resulting Kernel-PDEs are optimized using Sumof- Squares (SOS) decomposition and solved via semidefinite programming, with sufficient precision to guarantee the stability of the system in the L2-norm. This formulation allows optimizing extra degrees of freedom where the Kernel-PDEs are included as constraints. Uniqueness and invertibility of the Fredholm-type transformation are proved for polynomial Kernels in the space of continuous functions. The effectiveness and limitations of the approach proposed are illustrated by numerical solutions of some Kernel-PDEs

Solution of the Volterra-Fredholm integral equations via the Bernstein polynomials and least squares approach

We develop a numerical scheme to solve a general category of VolterraFredholm integral equations. For this purpose, the Bernstein polynomials and their features have been used. We convert the main equation into a set of algebraic equations in which the coefficient matrix is obtained by the least squares approximation approach. The error analysis is given to corroborate the precision of the proposed method. Numerical results are presented to demonstrate the success of the scheme for solving integral equations.Publisher's Versio

A new numerical technique based on Chelyshkov polynomials for solving two-dimensional stochastic It\^o-Volterra Fredholm integral equation

In this paper, a two-dimensional operational matrix method based on Chelyshkov polynomials is implemented to numerically solve the two-dimensional stochastic It\^o-Volterra Fredholm integral equations. These equations arise in several problems such as an exponential population growth model with several independent white noise sources. In this paper a new stochastic operational matrix has been derived first time ever by using Chelyshkov polynomials. After that, the operational matrices are used to transform the It\^o-Volterra Fredholm integral equation into a system of linear algebraic equations by using Newton cotes nodes as collocation point that can be easily solved. Furthermore, the convergence and error bound of the suggested method are well established. In order to illustrate the effectiveness, plausibility, reliability, and applicability of the existing technique, two typical examples have been presented

Solving Linear Volterra – Fredholm Integral Equation of the Second Type Using Linear Programming Method

في هذا البحث تم عرض تقنية جديدة لإيجاد حل لثلاثة أنواع من المعادلات التكاملية الخطية من النوع الثاني المتضمنة: معادلة فولتيرا-فريدهولم التكاملية (LVFIE) ( الحالة العامة), معادلة فولتيرا التكاملية (LVIE) و معادلة فريدهولم التكامليىة (LFIE) (كحالتين خاصتين). التقنية الجديدة تعتمد على تقريب الحل الى متعددة حدود من الدرجة (m-1) وبعد ذلك تحويل المسألة الى مسألة برمجة خطية (LPP) والتي سوف تحل لايجاد الحل التقريبي لمعادلة فولتيرا- فريدهولم التكاملية الخطية من النوع الثاني(LVFIE) . علاوة على ذلك تم استخدام الطرق التربيعية التي تضم: قاعدة شبه المنحرف (TR), قاعدة سمبسون 3/1 (SR), قاعدة بول (BR) وصيغة رومبرك للتكامل (RI) لتقريب التكامل الموجود في LVFIE , كما تم عمل مقارنات بين هذه الطرق. واخيرا لزيادة التوضيح تم اعطاء الخوارزمية المتبعة في الحل وتم تطبيقها على امثلة اختبارية لتوضيح فعالية التقنية الجديدة.In this paper, a new technique is offered for solving three types of linear integral equations of the 2nd kind including Volterra-Fredholm integral equations (LVFIE) (as a general case), Volterra integral equations (LVIE) and Fredholm integral equations (LFIE) (as special cases). The new technique depends on approximating the solution to a polynomial of degree  and therefore reducing the problem to a linear programming problem(LPP), which will be solved to find the approximate solution of LVFIE. Moreover, quadrature methods including trapezoidal rule (TR), Simpson 1/3 rule (SR), Boole rule (BR), and Romberg integration formula (RI) are used to approximate the integrals that exist in LVFIE. Also, a comparison between those methods is produced. Finally, for more explanation, an algorithm is proposed and applied for testing examples to illustrate the effectiveness of the new technique

Convergence analysis and parity conservation of a new form of a quadratic explicit spline with applications to integral equations

In this study, a new form of a quadratic spline is obtained, where the coefficients are determined explicitly by variational methods. Convergence is studied and parity conservation is demonstrated. Finally, the method is applied to solve integral equations.Fil: Ferrari, Alberto José. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Lara, Luis Pedro. Universidad del Centro Educativo Latinoamericano; ArgentinaFil: Santillan Marcus, Eduardo Adrian. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin

A Comparative Study Between ADM and MDM for a System of Volterra Integral Equation

In this paper, a comparative study between Adomain decomposition method (ADM) and Modified decomposition method (MDM) for a system of volterra integral equation. From the illustrate examples it is observed that the exact solution is smaller in both methods, the modified decomposition method is more proficient than its traditional ones it is less complicated, needs less time to get to the solution and most importantly the exact solution is achieved in two iterations