A new operational matrix method based on the Bernstein polynomials for solving the backward inverse heat conduction problems

Abstract

Purpose -The purpose of this paper is to introduce a novel approach based on the high-order matrix derivative of the Bernstein basis and collocation method and its employment to solve an interesting and ill-posed model in the heat conduction problems, homogeneous backward heat conduction problem (BHCP). Design/methodology/approach -By using the properties of the Bernstein polynomials the problems are reduced to an ill-conditioned linear system of equations. To overcome the unstability of the standard methods for solving the system of equations an efficient technique based on the Tikhonov regularization technique with GCV function method is used for solving the ill-condition system. Findings -The presented numerical results through table and figures demonstrate the validity and applicability and accuracy of the technique. Originality/value -A novel method based on the high-order matrix derivative of the Bernstein basis and collocation method is developed and well-used to obtain the numerical solutions of an interesting and ill-posed model in heat conduction problems, homogeneous BHCP with high accuracy.

An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs

“…with the initial conditions (16). Thus, e j,m (x) can be obtained by solving these sets of linear or nonlinear equations.…”

“…Maleknejad et al [15] proposed a numerical method for solving the systems of high order linear Volterra-Fredholm integro-differential equations by using Bernstein operational matrices. Rostamy and Karimi [16] presented a numerical method consists of the high-order derivative matrix of the Bernstein polynomials.…”

“…= 0, c e 0,2 = 0. The last row comes from(16). Therefore, the estimations of the errors are found asc e 0,1 = 0, c e 1,1 = −0.5769230763 × 10 −11 , c e 2,1 = 0.6615384615 × 10 −10 , c e 0,2 = 0, c e 1,2 = 0.3884615384 × 10 −10 , c e 2,2 = 0.8923076921 × 10 −10 , that is e 1,2 (x) e 2,2 (x) −0.1153846153 × 10 −10 x + 0.7769230768 × 10 −10 x 2 0.7769230768 × 10 −10 x + 0.1153846153 × 10 −10 x 2 .…”

Existence and Uniqueness of Solutions for Fractional Integro-Differential Equations and Their Numerical Solutions

“…The Bernstein polynomials have useful properties [19] like continuity and unity partition on interval [0, 1]. Recently, they have been applied to find numerical solution of various linear and nonlinear problems such as multi order fractional differential equations [14], Bessel differential equation [20], fractional Riccati type differential equations [21], Lane-Emden type equations [22], backward inverse heat condition problems [23], Volterra integral equations with convolution kernels [24] and variable order linear cable equation [25]. In this paper, we apply them to obtain numerical solution of the problem (1).…”

Approximate Solutions of Singular Differential Equations With Estimation Error by Using Bernstein Polynomials

“…Recently, Yiming Chen used Bernstein polynomials to find Numerical solution for the variable order linear cable equation [8]. Rostamy used a new operational matrix method based on the Bernstein polynomials for solving the backward inverse heat conduction problems [9].…”