scholarly journals Error Upper Bounds for a Computational Method for Nonlinear Boundary and Initial-Value Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Osvaldo Guimarães ◽  
José Roberto C. Piqueira
Keyword(s):  
Hilbert Space ◽  
Initial Value Problems ◽  
Nonlinear Boundary ◽  
Upper Bounds ◽  
Computational Approach ◽  
Computational Method ◽  
Operational Matrices ◽  
Initial Value ◽  
Accuracy Measures

This work develops a computational approach for boundary and initial-value problems by using operational matrices, in order to run an evolutive process in a Hilbert space. Besides, upper bounds for errors in the solutions and in their derivatives can be estimated providing accuracy measures.

Dual extremum principles for the heat equation

1977 ◽  
Vol 77 (3-4) ◽  
pp. 273-292 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Initial Value Problem ◽  
General Class ◽  
Initial Value Problems ◽  
Linear Operators ◽  
Point Boundary ◽  
Initial Value ◽  
Quantities Of Interest

SynopsisDual extremum principles characterising the solution of initial value problems for the heat equation are obtained by imbedding the problem in a two-point boundary-value problem for a system in which the original equation is coupled with its adjoint. Bounds on quantities of interest in the original initial value problem are obtained. Such principles are examples of ones which can be obtained for a general class of linear operators on a Hilbert space.

A New Estimate of the Sing Method for Linear Parabolic Problems Including the Initial Point

2004 ◽  
Vol 4 (2) ◽  
pp. 163-179 ◽  
Author(s):  
Ivan Gavrilyuk ◽  
Vladimir L. Makarov ◽  
Vitaliy Vasylyk
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Initial Value Problems ◽  
Initial Point ◽  
Parabolic Problems ◽  
Initial Value ◽  
Cauchy Integrals ◽  
Sinc Quadrature

Abstract For two new effcient methods for solving initial value problems in a Hilbert or Banach spaces based on a Sinc quadrature for an improper Dunford-Cauchy integrals over a path enveloping the spectrum of the operator we give a new unified estimate in the case of a Hilbert space.

Existence and uniqueness for two-point boundary value problems

1981 ◽  
Vol 88 (3-4) ◽  
pp. 219-234 ◽  
Author(s):  
John V. Baxley ◽  
Sarah E. Brown
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Shooting Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Conditions ◽  
Point Boundary ◽  
Initial Value

SynopsisBoundary value problems associated with y″ = f(x, y, y′) for 0 ≦ x ≦ 1 are considered. Using techniques based on the shooting method, conditions are given on f(x, y,y′) which guarantee the existence on [0, 1] of solutions of some initial value problems. Working within the class of such solutions, conditions are then given on nonlinear boundary conditions of the form g(y(0), y′(0)) = 0, h(y(0), y′(0), y(1), y′(1)) = 0 which guarantee the existence of a unique solution of the resulting boundary value problem.

scholarly journals Lucas polynomials semi-analytic solution for fractional multi-term initial value problems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Mahmoud M. Mokhtar ◽  
Amany S. Mohamed
Keyword(s):  
Initial Value Problems ◽  
Analytic Solution ◽  
Collocation Methods ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Differentiation ◽  
Initial Value ◽  
Lucas Polynomials ◽  
Error Analyses ◽  
Generalized Lucas Polynomials

AbstractHerein, we use the generalized Lucas polynomials to find an approximate numerical solution for fractional initial value problems (FIVPs). The method depends on the operational matrices for fractional differentiation and integration of generalized Lucas polynomials in the Caputo sense. We obtain these solutions using tau and collocation methods. We apply these methods by transforming the FIVP into systems of algebraic equations. The convergence and error analyses are discussed in detail. The applicability and efficiency of the method are tested and verified through numerical examples.

scholarly journals A decomposition algorithm coupled with operational matrices approach with applications to fractional differential equations

2021 ◽  
Vol 25 (Spec. issue 2) ◽  
pp. 449-455
Author(s):  
Imran Talib ◽  
Nur Alam ◽  
Dumitru Baleanu ◽  
Danish Zaidi
Keyword(s):  
Numerical Method ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Legendre Function ◽  
Operational Matrix ◽  
Decomposition Algorithm ◽  
Operational Matrices ◽  
Initial Value ◽  
Multiple Orders ◽  
Fractional Differential

In this article, we solve numerically the linear and non-linear fractional initial value problems of multiple orders by developing a numerical method that is based on the decomposition algorithm coupled with the operational matrices approach. By means of this, the fractional initial value problems of multiple orders are decomposed into a system of fractional initial value problems which are then solved by using the operational matrices approach. The efficiency and advantage of the developed numerical method are highlighted by comparing the results obtained otherwise in the literature. The construction of the new derivative operational matrix of fractional legendre function vectors in the Caputo sense is also a part of this research. As applications, we solve several fractional initial value problems of multiple orders. The numerical results are displayed in tables and plots.

scholarly journals A new computational algorithm for the solution of second order initial value problems in ordinary differential equations

2018 ◽  
Vol 3 (1) ◽  
pp. 167-174 ◽  
Author(s):  
P.K. Pandey
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Computational Algorithm ◽  
Second Order ◽  
Computational Method ◽  
Computationally Efficient ◽  
Initial Value ◽  
Computational Performance ◽  
Non Linear

AbstractIn this article, we propose a new computational method for second order initial value problems in ordinary differential equations. The algorithm developed is based on a local representation of theoretical solution of the second order initial value problem by a non-linear interpolating function. Numerical examples are solved to ensure the computational performance of the algorithm for both linear and non-linear initial value problems. From the results we obtained, the algorithm can be said computationally efficient and effective.

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