Finding a control parameter in a one-dimensional parabolic inverse problem by using the Bernstein Galerkin method

Abstract

An inverse problem concerning a diffusion equation with source control parameter is considered. The approximation of the problem is based on the Bernstein polynomial basis. The properties of Bernstein polynomials are first presented. The Bernstein polynomial basis vanishes except the first polynomial at x = 0, which is equal to 1 and the last polynomial at x = R, which is also equal to 1 over the interval [0, R]. This provides greater flexibility in which the boundary conditions are imposed at the end points of the interval. These properties together with the Galerkin method are then utilized to reduce the inverse problem to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Keywords:

• Bernstein polynomial
• Galerkin method
• inverse problem
• source control parameter
• parabolic partial differential equations

1. Introduction

The literature on the numerical approximation of solutions of parabolic partial differential equations with standard conditions is large and still growing rapidly. While many finite difference, finite element, spectral, finite volume and boundary element methods have been proposed to approximate such solutions, there has been much less research into the numerical approximation of parabolic partial differential equations with overspecified boundary data. Over the last few years, it has become increasingly apparent that many physical phenomena can be described in terms of parabolic partial differential equations with source control parameters. These types of equations appears in a variety of situations, for example, in the study of heat conduction processes, thermo-elasticity, chemical diffusion and control theory 1–8.

In this article we shall consider an inverse problem of finding a source parameter p(t) in the following diffusion equation: (1) with initial condition (2) and boundary conditions (3) subject to the overspecification at a point in the spatial domain (4) Where φ, f, g₀, g₁ and E are known functions, |E(t)| > 0, α and q are known constant, while the functions u and p are unknown. Certain types of physical problems can be modelled by (1)–(4) 1–16. Equation (1) can be used to describe a heat transfer process with a source parameter present. Equation (4) represents the temperature at a given point x*, in a spatial domain at time t. Thus, the purpose of solving this inverse problem is to identify the source parameter that will produce at each time t a desired temperature at a given point x* in a spatial domain. The existence and uniqueness of the solutions to this problem and also some more applications are discussed in 1–16.

In 1912, Bernstein 17 found the proof of the Weierstrass approximation theorem based on the law of large numbers for a sequence of Bernoulli trials. He constructed, for any continuous function f ∈ C[0, 1], a sequence of polynomials and proved that the sequence converges to f for n → ∞ uniformly with respect to x ∈ [0, 1]. These polynomials, called Bernstein polynomials, possess many remarkable properties. They have been studied intensively, and their connections with different branches of analysis, such as convex and numerical analysis, total positivity and the theory of monotone operators, have been investigated. Due to the fact that B_n(f; x) is an approximating sequence of shape-preserving operators, Bernstein polynomials play an important role in computer-aided geometric design 18. Basic facts on Bernstein polynomials, their generalizations, convergence and applications, can be found in, e.g. 17–24.

This article is organized as follows: In Section 2, we describe the basic formulation of Bernstein polynomials required for our subsequent development. Section 3 is devoted to the solution of Equation (1) by using the Bernstein Galerkin method. In Section 4, we report our numerical finding and demonstrate the accuracy of the proposed scheme by considering numerical examples. Section 5 consists of a brief summary.

2. Properties of the Bernstein polynomial

The general form of the Bernstein polynomials of m-th degree are defined on the interval [0, R] as 24: (5)

A recursive definition can also be used to generate the Bernstein polynomials over [0, R] so that the i-th m-th degree Bernstein polynomial can be written as: (6)

The Bernstein polynomial bases vanish except the first polynomial at x = 0, which is equal to 1 and the last polynomial at x = R, which is also equal to 1 over the interval [0, R]. This provides greater flexibility in which boundary conditions are impose at the end points of the interval. It can be readily shown that each of the Bernstein polynomials is positive and also the sum of all the Bernstein polynomials is unity for all real x belonging to the interval [0, R], that is, It can be easily shown that any given polynomial of degree m can be expanded in terms of linear combination of the basis functions (7)

2.1. Function approximation

A function f(t) defined over [0, R] may be approximated by the Bernstein polynomials as (8) where (9) (10) with (11) where are dual functions of B_i,m(t). These can be obtained by linear combinations of B_i,m(t), i = 0, 1, …, m, as follows.

Let be the dual functions of B given by (12)

We have (13) where I is (m + 1) × (m + 1) identity matrix. Let (14) then we have (15) Now a function u(x, t) of two independent variables defined for 0 ≤ x ≤ R₁ and 0 ≤ t ≤ R₂ may be approximated in terms of double Bernstein polynomials as (16) where K is a (m₁ + 1) × (m₂ + 1) matrix and (17)

3. Bernstein Galerkin method

Consider the inverse problem (18) with initial condition (19) and boundary conditions (20) subject to the overspecification at a point in the spatial domain (21) Let (22)

A Galerkin approximation to (22) is constructed as follows. The approximation u_NM and p_L is sought in the form of the truncated series (23) (24) where (25) and (26) Let (27)

ψ_k(t) = B_k,L(t),

where B_i,N(x), B_j,M(t) and B_k,L(t) are Bernstein polynomials.

In (23) w(x, t) is not unique. We can have different choice. If w(x, t) is chosen to be w(x, t) = h(x)g(t), then from Equation (26) we obtain (28)

In general case, w(x, t) can be chosen as interpolating function that satisfies the boundary conditions (3) and extra condition (4), that is, (29)

In the case of the expansion coefficients c_ij and p_k are determined by the Galerkin equations (30) where ⟨·⟩ denotes the inner product defined by (31) and k₁, k₂ are such that (k₁ + 1) × (k₂ + 1) = (N + M + 2) × (L + 1).

Galerkin Equations (30) gives a system of (N + M + 2) × (L + 1) non-linear equations which can be solved for the elements of c_ij, i = 0, 1, …, N, j = 0, 1, …, M and p_k, k = 0, 1, …, L using Newton's iterative method.

But in the general case of w(x, t), Equation (29), we add the collocation equations (32) to the Galerkin Equations (30). Suitable collocation points are zeroes of Chebyshev polynomials [25] (33)

4. Illustrative example

4.1. Example 1

Consider the equation (34) with initial conditions (35)

We applied the method presented in this article with M = N = 2, L = 2 and solved Equation (34).

From Equation (28) we have

and from Equation (30) we obtain

Thus from (23) and (24) we have

which is the exact solution.

4.2. Example 2

Consider the inverse problem (36) with initial conditions (37) By using the above-mentioned method with M = N = 2 and L = 1 we have

and from Equation (30) we obtain

Thus from (23) and (24) we have

which is the exact solution.

4.3. Example 3

Consider the equation (38) with initial conditions (39)

We applied the method presented in this article with M = N = 2, L = 2 and solved Equation (38). From Equation (28) we have

and from Equation (30) we obtain

Thus from (23) and (24) we have

which is the exact solution.

5. Conclusion

The properties of the Bernstein polynomials together with the Galerkin method are used to reduce the solution of the inverse problem to the solution of algebraic equations. The choice of basis and w(x, t) in the Bernstein Galerkin method provides greater flexibility in which to impose initial and boundary conditions. Moreover, only a small number of bases are needed to obtain a satisfactory result.

Illustrative examples are included to demonstrate the validity and applicability of the technique. It is also shown that the Bernstein Galerkin method provides an exact solution for the inverse problems. The given numerical examples support this claim.