Abstract
A complete family of solutions for the one-dimensional reaction-diffusion equation, , with a coefficient depending on is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper, the Dirichlet boundary conditions are considered; however, the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.
1. Introduction
In the present work a complete system of solutions of a one-dimensional reaction-diffusion equation with a variable coefficientconsidered on is obtained. We assume that the potential may be complex valued. The completeness of the system is with respect to the uniform norm in the closed rectangle . The system of solutions is shown to be useful for uniform approximation of solutions of initial boundary value problems for (1) as well as for explicit solution of the noncharacteristic Cauchy problem (see [1]) for (1) in terms of the formal powers arising in the spectral parameter power series (SPPS) method (see [2, 3]). In the paper, the Dirichlet boundary conditions are considered; however the proposed method can be easily extended onto other standard boundary conditions.
The complete system of solutions is constructed with the aid of the transmutation operators relating (1) to the heat equation (see, e.g., [4–6]). The possibility of constructing complete systems of solutions by means of transmutation operators was proposed and explored in [4], and the approach developed in [4] requires the knowledge of the transmutation operators. In the present work using a mapping property of the transmutation operators discovered in [7] we show that the construction of the complete systems of solutions for equations of form (1), representing transmuted heat polynomials, can be realized with no previous construction of the transmutation operator. Moreover, the use of the mapping property leads to an explicit solution of the noncharacteristic Cauchy problem for (1) with Cauchy data belonging to a Holmgren class [1].
We illustrate the implementation of the complete system of the transmuted heat polynomials by a numerical solution of an initial boundary value problem for (1). The approximate solution is sought in the form of a linear combination of the transmuted heat polynomials and the initial and boundary conditions are satisfied by a collocation method. A remarkable accuracy is achieved in few seconds using MATLAB 2012 on a usual PC.
Besides this Introduction the paper contains five sections. In Section 2 we recall the transmutation operators and some of their properties. In Section 3 an explicit solution of the noncharacteristic Cauchy problem for (1) is obtained. In Section 4 we prove the completeness of the transmuted heat polynomials. In Section 5 the numerical method for solving initial boundary value problems for (1) implementing the transmuted heat polynomials is discussed. Section 6 presents a numerical illustration.
2. Transmutation Operators and Formal Powers
2.1. System of Recurrent Integrals
Let be a continuous complex valued function defined on the segment . Throughout the paper we suppose that is a nonvanishing solution of the equationon such that and , where is a complex number. In [3] the existence of such solution was proved.
Consider two sequences of recurrent integrals (see [2, 3]):
Definition 1. The family of functions constructed according to the ruleis called the system of formal powers associated with .
The formal powers arise in the spectral parameter power series (SPPS) representation for solutions of the one-dimensional Schrödinger equation (see [2, 3]).
2.2. The Transmutation Operator
For any it is a well-known result [5, Chapter ] that there exists a function defined on the domain , continuously differentiable, such that the equalityis valid for all , where , , and has the form of a second-kind Volterra integral operator The operator is called transmutation operator. Moreover, the function is not unique and can be chosen so that (see, e.g., [7, 8]). When such function is the unique solution of the Goursat problem For any the kernel can be defined as and , with being the unique solution of the Goursat problem If the potential is times continuously differentiable on , the kernel is times continuously differentiable with respect to both independent variables.
The following mapping property of the operator is used throughout the paper.
Proposition 2 (see [7]).
The inverse operator also has the form of a second-kind Volterra integral operator and satisfies the following correspondence of the initial values (see [8]):where .
3. The Noncharacteristic Cauchy Problem for (1)
In this section an explicit solution of the noncharacteristic Cauchy problem for (1) in terms of the formal powers is obtained. This fact is a direct consequence of the mapping property (9).
Definition 3 (see [1]). For the positive constants , , and , the Holmgren class is the set of infinitely differentiable functions defined on that satisfyfor all .
Proposition 4. Let and be a solution of the noncharacteristic Cauchy problemwhere , . Then the series converges uniformly and absolutely for to the solution , where are the formal powers (4).
Remark 5. Note that the functions coincide with the formal powers (4) constructed starting with the particular solution of (2) satisfying the initial conditions and ; see [9, Proposition ].
Proof. Let be a solution of (12). Consider the function , where the operator is applied with respect to the variable . The function is a solution of the heat equation (cf. [4, Theorem ]) and due to (10) satisfies the following noncharacteristic Cauchy problem:Since , the solution of this problem is given by the absolutely and uniformly convergent series for (see, e.g., [1]): Due to (9) we obtain This series converges uniformly and absolutely for due to the uniform boundedness of and of its inverse.
4. Transmuted Heat Polynomials
In this section a complete system of solutions of (1) is presented.
Consider the heat polynomials (see, e.g., [10]) defined by Due to (9) we obtain that the functionsare solutions of (1) for all and . Indeed, we have that , , and
The completeness of the system of the heat polynomials with respect to the maximum norm proved in [4] and the uniform boundedness of and imply the completeness of (20) in the space of classical solutions of (1). Thus, the following statement is true.
Theorem 6. Let be continuous in and satisfy (1) in . Then given there exist and constants such that
Proof. Choose . Consider . Due to the completeness of (19), for any there exist and constants such that Then where the constant is the uniform norm of . The choice of finishes the proof.
5. Solution of Initial Boundary Value Problems for (1)
Consider the problem to find the solution of the equationsubject to the Dirichlet boundary conditionsand the initial conditionwhere , , and are continuously differentiable functions satisfying the compatibility conditions Problem (25)–(27) possesses a unique solution which depends continuously on the data (see, e.g., [11]).
The result of Theorem 6 suggests the following simple method to approximate the solution of problem (25)–(27). The approximate solution is sought in the formSince every is a solution of (25), their linear combination satisfies (25) as well. The coefficients are sought in such a way that satisfy the initial and the boundary conditions approximately. For this we used the collocation method. points are chosen on the parabolic boundary in order to construct a linear system of equations for the coefficients :System (30) is the result of imposing conditions (26) and (27) onto the approximate solution (29). Using the pseudoinverse matrix system (30) is solved, and the approximate solution (29) is computed on using the obtained coefficients and the definition of the transmuted heat polynomials (20).
Needless to add, the same approach is applicable to other kinds of boundary conditions.
6. Numerical Illustration
We present a numerical example of the application of the method described in the previous section. It reveals a remarkable accuracy with very little computational efforts. The implementation was realized in MATLAB 2012.
On the first step a nonvanishing solution of (2) was computed using the SPPS method (see [3]). The formal powers were constructed like in [6] using the spapi and fnint MATLAB routines from the spline Toolbox. Then the transmuted heat polynomials (20) were calculated. In order to obtain a unique solution of system (30) equally spaced points on the parabolic boundary were chosen. Finally, the approximate solution (29) was computed on a mesh of points in the interior of the rectangle and compared with the corresponding exact solution.
Example 1. Consider the initial Dirichlet problem
The exact solution of this problem has the form
The distribution of the absolute error of the approximate solution for is presented on Figure 1. The maximum absolute error of the approximate solution is of order .
It is often stated that boundary collocation methods (in particular, the heat polynomials method) lead to ill-conditioned systems of linear equations; see [12–14]. It is also the case for the proposed method. As is illustrated in Table 1, the condition number of the matrix in (30) grows rather fast. Nevertheless, the straightforward implementation of the proposed method presented no numerical difficulties. The convergence and the robustness of the method are illustrated in Table 1 where the maximum absolute and the maximum relative error of the approximate solution for different values of used for approximation (29) are presented. As one can appreciate, the convergence rate for small values of is exponential, while values of considerably larger than the optimum do not lead to significant loss of precision. Moreover, a simple test based on the accuracy of fulfilment of the initial and boundary conditions (32)-(33) can be utilized to estimate both the optimal and the accuracy of the obtained approximate solution.
7. Conclusions
A complete system of solutions of equation (1) is obtained. The solutions represent the images of the heat polynomials under the action of the transmutation operator. They are shown to be convenient for uniform approximation of solutions of initial boundary value problems for (1) as well as for explicit solution of the noncharacteristic Cauchy problem. Besides the Dirichlet boundary conditions considered in this paper the method is applicable to other standard boundary conditions. The complete system of solutions obtained can be used for solving moving and free boundary problems.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
Research was supported by CONACYT, Mexico, via Project 222478.