# Analytic Approximation of Solutions of Parabolic Partial Differential Equations with Variable Coefficients

Authors: Vladislav V. Kravchenko; Josafath A. Otero; Sergii M. Torba
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

A complete family of solutions for the one-dimensional reaction-diffusion equation,uxx(x,t)-q(x)u(x,t)=ut(x,t), with a coefficient q depending on x 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.


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## Body

## 1. Introduction

In the present work a complete system of solutions of a one-dimensional reaction-diffusion equation with a variable coefficient(1)uxxx,t-qxux,t=utx,tconsidered on Ω≔(-b,b)×(0,τ) is obtained. We assume that the potential q∈C[-b,b] may be complex valued. The completeness of the system is with respect to the uniform norm in the closed rectangle Ω¯. 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 Section2 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

Letq be a continuous complex valued function defined on the segment [-b,b]. Throughout the paper we suppose that f is a nonvanishing solution of the equation(2)f′′-qxf=0on (-b,b) such that f(0)=1 and f′(0)=α, where α is a complex number. In [3] the existence of such solution was proved.Consider two sequences of recurrent integrals (see [2, 3]):(3)X0≡1,Xnx=n∫0xXn-1sf2s-1nds,X~0≡1,X~nx=n∫0xX~n-1sf2s-1n-1ds,x∈-b,b,n∈N.Definition 1.
The family of functions{φk}k=0∞ constructed according to the rule(4)φkx=fxXkx,koddfxX~kx,kevenis called the system of formal powers associated with f.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 anyq∈C[-b,b] it is a well-known result [5, Chapter 1] that there exists a function (x,s)↦K(x,s) defined on the domain 0≤s≤|x|≤b, continuously differentiable, such that the equality(5)ATv=TBvis valid for all v∈C2[-b,b], where A≔∂2/∂x2-q, B≔∂2/∂x2, and T has the form of a second-kind Volterra integral operator (6)Tvx≔vx+∫-xxKx,svsds.The operator T is calledtransmutation operator. Moreover, the function K is not unique and can be chosen so that T[1]=f (see, e.g., [7, 8]). When q∈C1[-b,b] such function K is the unique solution of the Goursat problem (7)Kxxx,s-qxKx,s=Kssx,s,Kx,x=α2+12∫0xqydy,Kx,-x=α2,x∈-b,b.For any q∈C[-b,b] the kernel K can be defined as K(x,s)=Hx+s/2,x-s/2 and s≤x≤b, with H being the unique solution of the Goursat problem (8)Huvu,v=qu+vHu,v,Hu,0=α2+12∫0uqsds,H0,v=α2.If the potential q is n times continuously differentiable on (-b,b), the kernel K(x,s) is n+1 times continuously differentiable with respect to both independent variables.The following mapping property of the operatorT is used throughout the paper.Proposition 2 (see [7]).
(9)
T
x
k
=
φ
k
x
,
∀
k
∈
N
0
.The inverse operatorT-1 also has the form of a second-kind Volterra integral operator and satisfies the following correspondence of the initial values (see [8]):(10)v0=u0,v′0=u′0-αu0,where v≔T-1u.


## 2.1. System of Recurrent Integrals

Letq be a continuous complex valued function defined on the segment [-b,b]. Throughout the paper we suppose that f is a nonvanishing solution of the equation(2)f′′-qxf=0on (-b,b) such that f(0)=1 and f′(0)=α, where α is a complex number. In [3] the existence of such solution was proved.Consider two sequences of recurrent integrals (see [2, 3]):(3)X0≡1,Xnx=n∫0xXn-1sf2s-1nds,X~0≡1,X~nx=n∫0xX~n-1sf2s-1n-1ds,x∈-b,b,n∈N.Definition 1.
The family of functions{φk}k=0∞ constructed according to the rule(4)φkx=fxXkx,koddfxX~kx,kevenis called the system of formal powers associated with f.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 anyq∈C[-b,b] it is a well-known result [5, Chapter 1] that there exists a function (x,s)↦K(x,s) defined on the domain 0≤s≤|x|≤b, continuously differentiable, such that the equality(5)ATv=TBvis valid for all v∈C2[-b,b], where A≔∂2/∂x2-q, B≔∂2/∂x2, and T has the form of a second-kind Volterra integral operator (6)Tvx≔vx+∫-xxKx,svsds.The operator T is calledtransmutation operator. Moreover, the function K is not unique and can be chosen so that T[1]=f (see, e.g., [7, 8]). When q∈C1[-b,b] such function K is the unique solution of the Goursat problem (7)Kxxx,s-qxKx,s=Kssx,s,Kx,x=α2+12∫0xqydy,Kx,-x=α2,x∈-b,b.For any q∈C[-b,b] the kernel K can be defined as K(x,s)=Hx+s/2,x-s/2 and s≤x≤b, with H being the unique solution of the Goursat problem (8)Huvu,v=qu+vHu,v,Hu,0=α2+12∫0uqsds,H0,v=α2.If the potential q is n times continuously differentiable on (-b,b), the kernel K(x,s) is n+1 times continuously differentiable with respect to both independent variables.The following mapping property of the operatorT is used throughout the paper.Proposition 2 (see [7]).
(9)
T
x
k
=
φ
k
x
,
∀
k
∈
N
0
.The inverse operatorT-1 also has the form of a second-kind Volterra integral operator and satisfies the following correspondence of the initial values (see [8]):(10)v0=u0,v′0=u′0-αu0,where v≔T-1u.


## 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 φk is obtained. This fact is a direct consequence of the mapping property (9).Definition 3 (see [1]).
For the positive constantsγ1, γ2, and C1, the Holmgren class H(γ1,γ2,C1,t0) is the set of infinitely differentiable functions v defined on t-t0<γ2 that satisfy(11)vjt≤C1γ1-2j2j!,j=0,1,…for all t∈t-t0<γ2.Proposition 4.
Letq∈C[-b,b] and u(x,t) be a solution of the noncharacteristic Cauchy problem(12)uxxx,t-qxux,t=utx,t,-b<x<b,t<τ(13)u0,t=Ft,t<τ(14)ux0,t=Gt,t<τwhere F,G∈H(b,τ,C,0), C>0. Then the series (15)∑j=0∞Fjt2j!φ2jx-α2j+1φ2j+1x+Gjt2j+1!φ2j+1xconverges uniformly and absolutely for x≤r<b to the solution u(x,t), where φk are the formal powers (4).Remark 5.
Note that the functionsφ2j(x)-α/2j+1φ2j+1(x) coincide with the formal powers (4) constructed starting with the particular solution g of (2) satisfying the initial conditions g(0)=1 and g′(0)=0; see [9, Proposition 4.7].Proof.
Letu(x,t) be a solution of (12). Consider the function h≔T-1u, where the operator T-1 is applied with respect to the variable x. The function h is a solution of the heat equation (cf. [4, Theorem 2.1.2]) and due to (10) satisfies the following noncharacteristic Cauchy problem:(16)hxx=ht,-b<x<b,t<τh0,t=Ft,t<τhx0,t=Gt-αFt,t<τ.Since G-αF∈H(b,τ,(1+α)C,0), the solution of this problem is given by the absolutely and uniformly convergent series for x≤r<b (see, e.g., [1]): (17)hx,t=∑k=0∞Fkt2k!x2k+Gkt-αFkt2k+1!x2k+1.Due to (9) we obtain (18)ux,t=Thx,t=∑k=0∞Fkt2k!φ2kx-α2k+1φ2k+1x+Gkt2k+1!φ2k+1x.This series converges uniformly and absolutely for x≤r<b due to the uniform boundedness of T 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 (19)hnx,t=n!∑k=0n/2tkxn-2kk!n-2k!,n∈N0.Due to (9) we obtain that the functions(20)unx,t=n!∑k=0n/2tkφn-2kxk!n-2k!,n∈N0are solutions of (1) for all -b<x<b and t>0. Indeed, we have that un=Thn, n∈N0, and (21)∂2un∂x2-qxunx,t-∂un∂tx,t=T∂2hn∂x2-∂hn∂tx,t=0.The completeness of the system of the heat polynomials with respect to the maximum norm proved in [4] and the uniform boundedness of T and T-1 imply the completeness of (20) in the space of classical solutions of (1). Thus, the following statement is true.Theorem 6.
Letu(x,t) be continuous in Ω¯ and satisfy (1) in Ω. Then given ε>0 there exist N∈N and constants a0,a1,…,aN such that (22)maxΩ¯⁡ux,t-∑n=0Nanunx,t<ε.Proof.
Chooseε>0. Consider h(x,t)=T-1u(x,t). Due to the completeness of (19), for any ε1>0 there exist N∈N and constants a0,a1,…,aN such that (23)maxΩ¯⁡hx,t-∑n=0Nanhnx,t<ε1.Then (24)maxΩ¯⁡ux,t-∑n=0Nanunx,t=maxΩ¯⁡Thx,t-∑n=0NanThnx,t≤Cε1,where the constant C is the uniform norm of T. The choice of ε1=ε/C finishes the proof.


## 5. Solution of Initial Boundary Value Problems for (1)

Consider the problem to find the solution of the equation(25)uxxx,t-qxux,t=utx,t,x,t∈Ωsubject to the Dirichlet boundary conditions(26)u-b,t=ψ1t,ub,t=ψ2t,t∈0,τand the initial condition(27)ux,0=φx,x∈-b,b,where ψ1, ψ2, and φ are continuously differentiable functions satisfying the compatibility conditions (28)ψ10=φ-b,ψ20=φb.Problem (25)–(27) possesses a unique solution which depends continuously on the data (see, e.g., [11]).The result of Theorem6 suggests the following simple method to approximate the solution of problem (25)–(27). The approximate solution u~ is sought in the form(29)u~x,t=∑n=0Nanunx,t.Since every un is a solution of (25), their linear combination satisfies (25) as well. The coefficients ann=0N are sought in such a way that u~ satisfy the initial and the boundary conditions approximately. For this we used the collocation method. M points {(xi,ti)}i=1M are chosen on the parabolic boundary -b×0,τ∪-b,b×0∪b×0,τ in order to construct a linear system of equations for the coefficients ann=0N:(30)∑n=0Nanunxi,ti=ψ1ti,xi=-bφxi,ti=0ψ2ti,xi=b,i=1,…,M.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 Ω using the obtained coefficients ann=0N 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 solutionf of (2) was computed using the SPPS method (see [3]). The formal powers φk 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) N+1 equally spaced points on the parabolic boundary were chosen. Finally, the approximate solution (29) was computed on a mesh of 200×100 points in the interior of the rectangle and compared with the corresponding exact solution.Example 1.
Consider the initial Dirichlet problem(31)uxxx,t-x2ux,t=utx,t,x,t∈-1,1×0,1,(32)ux,0=e-0.5x2,x∈-1,1,(33)u-1,t=u1,t=e-0.5-t,t∈0,1.The exact solution of this problem has the form(34)ux,t=exp⁡-12x2-t.The distribution of the absolute error of the approximate solution forN=26 is presented on Figure 1. The maximum absolute error of the approximate solution is of order 10-13.Figure 1
The absolute value of the differenceux,t-u26x,t between the exact and the approximate solutions for problem (31)–(33).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 N used for approximation (29) are presented. As one can appreciate, the convergence rate for small values of N is exponential, while values of N 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 N and the accuracy of the obtained approximate solution.Table 1
Maximal absolute and relative errors of the approximate solution and condition number of the matrix in (30) for problem (31)–(33) obtained for different values of N in (29).
N
Max. absolute error
Max. relative error
Cond. number
5
2.3⋅10-2
6.2⋅10-2
55.7
10
1.4⋅10-4
5.5⋅10-4
1.82
⋅
1
0
5
15
9.6⋅10-8
4.0⋅10-7
3.65
⋅
1
0
9
20
2.0⋅10-10
8.6⋅10-10
1.76
⋅
1
0
14
23
7.6⋅10-13
3.2⋅10-12
1.67
⋅
1
0
17
26
1.8⋅10-13
7.8⋅10-13
2.59
⋅
1
0
23
29
2.5⋅10-12
1.1⋅10-11
1.23
⋅
1
0
23
34
1.7⋅10-10
7.3⋅10-10
1.37
⋅
1
0
25
39
2.3⋅10-9
9.8⋅10-9
2.86
⋅
1
0
29
50
4.7⋅10-10
2.1⋅10-9
1.25
⋅
1
0
41
75
6.1⋅10-11
2.7⋅10-10
2.89
⋅
1
0
73
100
2.8⋅10-10
1.2⋅10-9
5.63
⋅
1
0
105


## 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.



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