Advertisement

From Backward Approximations to Lagrange Polynomials in Discrete Advection–Reaction Operators

  • Francisco J. Solis
  • Ignacio Barradas
  • Daniel Juarez
Original Research

Abstract

In this work we introduce a family of operators called discrete advection–reaction operators. These operators are important on their own right and can be used to efficiently analyze the asymptotic behavior of a finite differences discretization of variable coefficient advection–reaction–diffusion partial differential equations. They consists of linear bidimensional discrete dynamical systems defined in the space of real sequences. We calculate explicitly their asymptotic evolution by means of a matrix representation. Finally, we include the special case of matrices with different eigenvalues to show the connection between the operators evolution and interpolation theory.

Keywords

Backward approximation Advection–reaction operators Infinite matrix Lagrange polynomials 

Introduction

The search for numerical methods for approximations for partial differential equations is an important and active area of research, see for example [6, 13, 23]. Among them, the backward approximations for the derivatives in a partial differential equation for a class of nonlinear functions give a well-posed discrete dynamical system [8, 9, 15]. Our motivation is derivated by several models from Biology, in particular from models of the evolution of infected cells by the Papilloma virus [10, 17, 18], where finite difference formulations to approximate the solution of several advection–reaction–diffusion equations in mesh were developed. In previous works a family of discrete delay advection–reaction operators was introduced along with an infinite matrix formulation in order to investigate the asymptotic behavior of the orbits of their iterates [19, 20, 21]. The infinite matrices obtained were triangular matrices with only one nonzero subdiagonal [19]. In the present work we generalize the previous framework to include infinite lower triangular matrices with two nonzero subdiagonals. Despite of the cumbersome calculations our goal is to obtain closed formulas that allow us to reduce the discretizations of variable coefficient advection-reaction-diffusion partial differential equations via finite differences to the evaluation of explicit expressions. It is important to remark that the proposed numerical scheme and its closed formulas may be useful for dealing also with random advection diffusion equations, since errors in the measurements and inherent uncertainties often involved in real phenomena motivate to consider those inputs as random variables or stochastic processes. For such equations an appropriate choice of a numerical scheme is crucial to compute reliable approximations of the expectation and variance of the solution of the corresponding stochastic process, see [4, 5, 11]. To achieve our goal, we present in Sect. 2 the problem of interest and introduce the discrete advection-reaction operators along with their matrix representation. In Sect. 3, we generalize the problem to include a more broader set of linear problems that can be solved with the same approach. In Sect. 4, we define proper polynomials in order to capture the mixing of indexes of the iterates of the infinite matrices. These polynomials are studied and some of their properties are obtained. It is in this section that we obtain the main formula for the powers of the matrices. Finally, in Sect. 5 we include the special case of matrices with different eigenvalues to show the connection between the operators evolution and interpolation theory. For completeness we illustrate the application of our results to several numerical examples.

Problem Formulation

In this section we seek a discrete linear operator related with the linear advection–reaction–diffusion equation
$$\begin{aligned} \frac{\partial u (x,t)}{\partial t}+\alpha _1(x)\frac{\partial u (x,t)}{\partial x} + \alpha _2(x)\frac{\partial ^2 u (x,t)}{\partial x^2}=\beta (x)u(x,t). \end{aligned}$$
(1)
By discretizing such equation with a backwards finite differences scheme we obtain
$$\begin{aligned} \frac{ u_{i}^{j+1}-u_{i}^{j} }{\Delta t}+ \alpha _1 \left( x \right) \frac{ u_{i}^{j}-u_{i-1}^{j} }{\Delta x}+ \alpha _2 \left( x \right) \frac{ u_{i-2}^{j}-2u_{i-1}^{j}+u_{i}^{j} }{\Delta x^2} = \beta u_{i}^{j}, \end{aligned}$$
(2)
where the subindex are related to the spatial variable and the superindex are related to the temporal variable, with \(\Delta x\) and \(\Delta t\) their respective step sizes. Equation (2) can be written as:
$$\begin{aligned} u_{i-2}^{j} \left( -\frac{\alpha _2\Delta t }{\Delta ^2 x} \right) + u_{i-1}^{j} \left( \frac{\alpha _1 \Delta t}{\Delta x}-\frac{2 \alpha _2\Delta t}{\Delta ^2 x} \right) + u_{i}^{j} \left( 1-\frac{\alpha _1\Delta t }{\Delta x}-\frac{\alpha _2\Delta t }{\Delta ^2x}+\beta \Delta t \right) =u_{i}^{j+1}. \end{aligned}$$
(3)
which is an infinite bidimensional discrete dynamical system. This leads to the following definition:

Definition

Given three real sequences \(\varvec{a}= \left\{ a_n \right\} _{n=1}^{\infty }\), \(\varvec{b}=\left\{ b_n \right\} _{n=1}^{\infty }\) and \(\varvec{\lambda }=\left\{ \lambda _n \right\} _{n=1}^{\infty }\), we define the bidimensional discrete dynamical system
$$\begin{aligned} u_{m,{n+1}}= \left\{ \begin{array}{ll} \lambda _m u_{m,n}&{} \quad \text{ if } n \in \mathbb {N} \text{ and } m=1\\ a_1u_{m-1,n}+\lambda _m u_{m,n}&{} \quad \text{ if } n \in \mathbb {N} \text{ and } m=2\\ &{} \\ b_{m-2} u_{m-2,n} +a_{m-1} u_{m-1,n} +\lambda _m u_{m,n} &{} \quad \text{ if } m,n \in \mathbb {N} \text{ and } m \ge 3. \end{array} \right. \end{aligned}$$
(4)

In order to relate the general dynamical system (4) with the discretization of equation (1), we choose \(a_n\), \(b_n\) and \(\lambda _n\) as \(a_{n}= \frac{ \alpha _1 \left( x_n \right) \Delta t }{\Delta x}-\frac{2 \alpha _2\left( x_n \right) \Delta t }{\Delta ^2x},\) \(b_{n}= -\frac{ \alpha _2 \left( x_n \right) \Delta t }{\Delta ^2x}\) and \( \lambda _n= 1-\frac{ \alpha _1\left( x_n \right) \Delta t}{\Delta x}-\frac{ \alpha _2 \left( x_n \right) \Delta t}{\Delta ^2x}+\beta \left( x_n \right) \Delta t \) for all \(n \in \mathbb {N}.\) Therefore, the discretization of (1) is a particular case of (4). It is important to mention that the analysis of two-dimensional discrete dynamical systems is one of the main current research frontiers and of great interest in applications, see for example [1, 2, 7, 12, 14, 16].

System (4) can be written in matrix form as
$$\begin{aligned} \varvec{u}_{n+1} = A_{\varvec{a},\varvec{b},\varvec{\lambda }} \varvec{u}_{n}, \end{aligned}$$
(5)
where \( A_{\varvec{a},\varvec{b},\varvec{\lambda }} \) is the following infinite lower triangular 3-band matrix
$$\begin{aligned} A_{\varvec{a},\varvec{b},\varvec{\lambda }}=\left( \begin{matrix} \lambda _1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \ldots \\ a_1 &{}\quad \lambda _2 &{}\quad 0 &{}\quad 0 &{}\quad \ldots \\ b_1 &{}\quad a_2 &{}\quad \lambda _3 &{}\quad 0 &{}\quad \ldots \\ 0 &{}\quad b_2 &{}\quad a_3 &{}\quad \lambda _4 &{}\quad \ldots \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots \\ \end{matrix}\right) \end{aligned}$$
(6)
where \(\varvec{u}_n\) is the infinite vector (a sequence) with entries \(u_{m,n}, m \in \mathbb {N}.\) Let \(\mathfrak {s}\) be the vector space of all real sequences and if \(\varvec{s} \in \mathfrak {s}\), we denote \(\varvec{s}=\left\{ s_n \right\} _{n=0}^{\infty }\). Then, matrix (6) defines a linear operator on the set of sequences, that is,
$$\begin{aligned} A_{\varvec{a},\varvec{b},\varvec{\lambda }}: \mathfrak {s} \rightarrow \mathfrak {s}, \end{aligned}$$
which will be called the discrete advection-reaction operator, (DAR operator), for \(\varvec{a}\), \(\varvec{b}\) and \(\varvec{\lambda }\).
The orbit of a given initial condition \(\varvec{s}_0 \in \mathfrak {s}\) for system (5) is given by the sequence
$$\begin{aligned} \small { \left\{ \varvec{s}_0, A_{\varvec{a},\varvec{b},\varvec{\lambda }} \varvec{s}_0, A_{\varvec{a},\varvec{b}, \varvec{\lambda }}^2 \varvec{s}_0, \ldots \right\} }, \end{aligned}$$
which gives the time evolution of \(\varvec{s}_0\). Thus, in order to determine the dynamics of (5) it is only necessary to calculate the powers of the matrix \(A_{\varvec{a},\varvec{b},\varvec{\lambda }}.\) From here on, we will fix the sequences \( \varvec{a}, \varvec{b}\) and \( \varvec{\lambda } \in \mathfrak { s } \) and, for simplicity of notation, denote the matrix \( A_ {\varvec{a}, \varvec{b}, \varvec{\lambda } } \) by A. Before we proceed to solve problem (5) it is worth mentioning that its solution actually covers a bigger variety of more general problems, which we discuss in the next section.

Problem Generalization

Although Problem (5) is equivalent to a great variety of more general problems, it is very difficult to recognize a matrix that can be brought back to the form (6). So, instead of looking for all similarity transformations, \(A=BCB^{-1}\) that take \(\varvec{u}_{n+1} = A \varvec{u}_{n}\) into \(\varvec{v}_{n+1} = C \varvec{v}_{n},\) where \(\varvec{v}_{n} = B^{-1} \varvec{u}_{n},\) we will look for matrices that can be easily recognized to be susceptible to be changed into form (6). Without loss of generality we assume that both sequences \(\varvec{a}\) and \( \varvec{b}\) have all nonzero entries. Those matrices will be characterize by conditions 1, 2, 3 and 4 given below and they consist in permutations of indexes of the original variables.

Consider an infinite matrix \(C=\{c_{i,j} \}.\) For convenience, every time we say that four nonzero entries are the vertices of a rectangle we mean that there are two pairs of such entries each in the same row, and two pairs of such entries each in the same column.

We assume that \(C=\{c_{i,j} \}\) satisfies the following conditions:
  1. 1.

    (Every column has exactly three nonzero entries). For every \(j \in \mathbb {N}\) there are exactly three natural numbers \(i_1,i_2,i_3\) such that \(c_{ij}=0\) for \(i \ne i_1,i_2,i_3\) and \(c_{ij} \ne 0\) for \(i = i_1,i_2,i_3.\)

     
  2. 2.
    (There are two special rows, one with exactly one nonzero entry and the second one with exactly two nonzero entries). There exist \(i_1,i_2,k_1\) and \(k_2\) such that
    1. (a)

      \(c_{i_1,k_1} \ne 0\) and \(c_{i,j} =0 \quad \) for all \(j \ne k_1,\) and

       
    2. (b)

      \( c_{i_2,k_1} \ne 0 \ne c_{i_2,k_2}\) and \(c_{i_2,j}=0\)    for all \(j \ne k_1,k_2.\)

       
     
  3. 3.

    (Every row different from the special two rows has exactly three nonzero entries).

     
  4. 4.

    (Every row, except one of the special ones defined on 2, is connected to a predecessor and to a successor) Given \(l \in \mathbb {N},\) different from \(i_1,i_2\) we denote \(k_1(l),k_2(l)\) and \(k_3(l)\) those entries such that \(c_{lk_j(l)} \ne 0\) for \(j=1,2,3.\)

    One of the entries \(c_{l,k_j(l)}\) satisfies (without loss in generality, we assume that \(c_{l,k_1(l)}\) is such entry ):
    1. (a)

      \(c_{l,k_1(l)}\) is a vertex of two different rectangles, one including \(c_{l,k_2(l)}\) and the other one including \(c_{l,k_3(l)}.\)

       
    2. (b)

      \(c_{l,k_2(l)}\) is a vertice of only one rectangle (likewise for \(c_{l,k_3(l)}\)).

       
     
Then the problem \(\hat{v}_{n+1} = C \hat{v}_{n} \) can be transformed by a permutation of indexes of the variables into a problem of the form (4).
In order to determine the permutation of the indexes, the following algorithm can be applied:
  1. (a)

    Find both special rows, the first one, denoted by \(i_1,\) with a nonzero entry; and the second one, denoted by \(i_2,\) with two nonzero entries.

     
  2. (b)

    To find the third row, denoted by \(i_3\) we look for the entry on column \(k_1\) not in row \(i_1\) and \(i_2\) (condition 2), its row will be \(i_3.\)

     
  3. (c)

    By condition (3) the entry \(c_{i_3,k_2} \ne 0;\) look for the third nonzero entry in column \(k_2\), its row will be \(i_4\)

     
  4. d)

    Repeat the two previous steps

     
One system that illustrates the generalization just described is given by the following example. Consider \(\hat{v}_{n+1} = C \hat{v}_{n} \) which is equivalent to problem \(\varvec{u}_{n+1} = A \varvec{u}_{n}\) via a variable permutation, where
$$\begin{aligned} C=\left( \begin{matrix} b_1 &{}\quad a_2 &{}\quad \lambda _3 &{}\quad 0 &{}\quad 0 &{}\quad \ldots \\ 0 &{}\quad \lambda _2 &{}\quad a_1 &{}\quad 0 &{}\quad 0 &{}\quad \ldots \\ 0 &{}\quad 0 &{}\quad \lambda _1 &{}\quad 0 &{}\quad \vdots &{}\quad \vdots \\ a_3 &{}\quad b_2 &{}\quad 0 &{}\quad \lambda _4 &{}\quad 0 &{}\quad \vdots \\ b_3 &{}\quad 0 &{}\quad 0 &{}\quad a_4 &{}\quad \lambda _5 &{}\quad \ddots \\ 0&{}\quad 0 &{}\quad 0 &{}\quad b_4 &{}\quad a_5 &{}\quad \lambda _6 \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots \\ \end{matrix}\right) \text{ and } A=\left( \begin{matrix} \lambda _1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \ldots \\ a_1 &{}\quad \lambda _2 &{}\quad 0 &{}\quad 0 &{}\quad \ldots \\ b_1 &{}\quad a_2 &{}\quad \lambda _3 &{}\quad 0 &{}\quad \ldots \\ 0 &{}\quad b_2 &{}\quad a_3 &{}\quad \lambda _4 &{}\quad \ldots \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots \\ \end{matrix}\right) . \end{aligned}$$
(7)
Notice that matrix C satisfies the conditions 1–4, with the special rows given by the second and third rows. Condition 4 for the first and fourth row, for example, is satisfied with \(c_{1,k_1(1)}=a_{2},\) \(c_{4,k_1(4)}=a_{3}.\)

Powers of A

Before obtaining an analytic expression for the powers of A we define three classes of unitary1 polynomials in the variables \(\lambda _j,\lambda _{j+1},...,\lambda _i\) with \(i>j.\) Such polynomials will be used to obtain a closed analytical expression for \(A^n.\)

Polynomials P, Q and R.

Definition

Given natural numbers ji and n,  with \(j< i\) and \(\alpha \) a multi-index, that is, an \((i-j)\)-tuple of integers \(\alpha _k\) (\(k=j,j+1,\dots ,i\)) with \(|\alpha |=\sum _{k=j}^i \alpha _k,\) we define
$$\begin{aligned}&Q \left[ \lambda _j,\lambda _i \right] ^n= \sum \limits _{|\alpha |=n} \lambda _j^{\alpha _j}\lambda _{j+1}^{\alpha _{j+1}}\ldots \lambda _{i-1}^{\alpha _{i-1}}\lambda _i^{\alpha _i}, \end{aligned}$$
(8)
$$\begin{aligned}&P \left[ \lambda _j,\lambda _i\right] ^n= \left\{ \begin{array}{ll} \sum _{|\alpha |=n} \lambda _j^{\alpha _j}\lambda _{j+2}^{\alpha _{j+2}}\ldots \lambda _{i-2}^{\alpha _{i-2}}\lambda _i^{\alpha _i}, &{} \quad \text{ if } i-j \text{ is } \text{ even } \\ 0, &{} \quad \text{ if } i-j \text{ is } \text{ odd. } \end{array} \right. \end{aligned}$$
(9)
Also, given natural numbers \(i_1,i_2,\dots ,i_m\) with \(j\le i_1<i_2<\dots <i_m\le i \) we define
$$\begin{aligned} R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^n= \sum _{|\alpha |=n} \lambda _j ^{\alpha _j} \ldots {\lambda } _{i_1}^{\alpha _{i_1}} \ldots {\lambda } _{i_m}^{\alpha _{i_m}} \ldots \lambda _i^{\alpha _i}, \end{aligned}$$
(10)
we set \({\alpha _{i_1}}=\cdots =\alpha _{i_m}=0\) in (10) in order to exclude the terms \(\lambda _{i_k}^{\alpha _{i_k}},\) \(k=1,2,3,..,m\) in the products.
Notice that
  • \( \ Q \left[ \lambda _j,\lambda _i \right] ^0 = \ P \left[ \lambda _j,\lambda _i\right] ^0 = \ R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^0 = 1.\)

  • If \(n<0\), then \( \ Q \left[ \lambda _j,\lambda _i \right] ^n = \ P \left[ \lambda _j,\lambda _i\right] ^n = \ R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^n = 0.\)

  • Polynomials Q and P can be seen as special cases of some polynomial R. They deserve a special name because they have a particular meaning in the following results.

  • \( Q \left[ \lambda _j,\lambda _{i-1} \right] ^n = R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i} \right\} \right] ^n \ \),

  • \( P \left[ \lambda _j,\lambda _i\right] ^n = \ R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{j+1},\lambda _{j+3},\ldots ,\lambda _{i-3},\lambda _{i-1} \right\} \right] ^n\)

Lemma 1

Polynomials Q, P and R satisfy the properties
$$\begin{aligned} Q \left[ \lambda _j,\lambda _i \right] ^n= & {} Q \left[ \lambda _j,\lambda _{i-1} \right] ^n+\lambda _i Q \left[ \lambda _j,\lambda _i \right] ^{n-1} \end{aligned}$$
(11)
$$\begin{aligned} P \left[ \lambda _j,\lambda _i \right] ^n= & {} P \left[ \lambda _j,\lambda _{i-2} \right] ^n+\lambda _i P \left[ \lambda _j,\lambda _i \right] ^{n-1} \end{aligned}$$
(12)
$$\begin{aligned} R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^n= & {} R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^n \nonumber \\&+\,\lambda _i R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^{n-1}. \end{aligned}$$
(13)

Proof

We only show the proof for the polynomials R since the other ones are analogous.
$$\begin{aligned}&\, R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^n \\&\quad = \sum _{r=0}^n \lambda _i^r R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^{n-r}\\&\quad = R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^n+ \sum _{r=1}^n \lambda _i^r R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^{n-r}\\&\quad = R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^n+ \lambda _i \sum _{r=1}^n \lambda _i^{r-1} R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^{n-r}\\&\quad = R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^n+ \lambda _i \sum _{r=0}^{n-1} \lambda _i^{r} R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^{n-r-1}\\&\quad = R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^n+ \lambda _i R \left[ \lambda _j,\lambda _{i},\left\{ \lambda _{i_1},\ldots ,\lambda _{i_m} \right\} \right] ^{n-1}. \end{aligned}$$
\(\square \)

Remark

The number of terms of each polynomial Q, R, P is given next. Let \(S \in \left\{ P,Q,R \right\} \) and n the degree of the polynomial under consideration, then the number of terms of the polynomial S of degree n, denoted by \(T_S\) is given by
$$\begin{aligned} T_S= \frac{ \left( N_S+n-1 \right) ! }{\left( N_S-1 \right) ! \text { }n!}, \end{aligned}$$
with \(N_P=\frac{i-j}{2}+1\), \(N_Q=i-j+1\), \(N_R=i-j+1-n_R\) where \(n_R\) denotes the number of variables omitted in the polynomial R.

Powers of A

Using the notation from the previous section we state the main result for the powers of A.

Theorem 4.1

Let A be an infinite band matrix as in (6), and \(\hat{a}\), \(\hat{b}\) and \(\hat{\lambda } \in \varvec{s}\). Then the ij entry of the nth power of \(A=\left\{ c_{i,j} \right\} \), denoted by \(c^n_{i,j}\), is given by
$$\begin{aligned} \left\{ \begin{array}{ll} 0, &{} \text{ if } i < j \\ \lambda _i^n &{} \text{ if } i=j\\ \left( \prod \limits _{k=j}^{i-1} a_k \right) Q \left[ \lambda _j,\lambda _i \right] ^{n-i+j} + \left( \prod \limits _{k=0}^{q-1} b_{j+2k} \right) P \left[ \lambda _j,\lambda _i \right] ^{n-q}+ &{} \text{ if } i>j \\ \left( \prod \limits _{k=j}^{i-1} a_k \right) \sum \limits _{s=1}^{\lceil q-1 \rceil } \mathop {\sum \limits _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \left( \prod \limits _{k=1}^{s} \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} \right) R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{n-i+j+s} &{} \end{array}\right. \end{aligned}$$
(14)
where \(q=\frac{i-j}{2}\) and \(i_0=j-2\).

Proof

We have three different cases (a) \(i<j\), (b) \(i=j\) and (c) \(i>j.\)
  1. (a)

    Since A is a lower triangular matrix then \(c^n_{i,j}=0\).

     
  2. (b)

    It is straightforward to show that \(c^n_{i,j}=\lambda _i^n\).

     
  3. (c)

    This case will be shown by induction on n.

     
When \(i>j\) and \(n=1\), we have to prove the following equality:
$$\begin{aligned} c^1_{i,j}= & {} \left( \prod _{k=j}^{i-1} a_k \right) Q \left[ \lambda _j,\lambda _i \right] ^{1-i+j} + \left( \prod _{k=0}^{q-1} b_{j+2k} \right) P \left[ \lambda _j,\lambda _i \right] ^{1-q} \\&+ \prod _{k=j}^{i-1} a_k \sum _{s=1}^{\lceil q-1 \rceil } \mathop {\sum _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{1-i+j+s}. \end{aligned}$$
To simplify the right hand side of the previous expression, let us check the conditions for which one of the three polynomials has a nonnegative integer exponent. For the polynomial Q we require that \(1-i+j \ge 0,\) \(j+1 \ge i>j,\) \(i=j+1,\) and \(q=\frac{i-j}{2}=\frac{1}{2}.\) Since q is not an integer \(P=0\) and \(\lceil q-1 \rceil =0\),
$$\begin{aligned} c^1_{j+1,j}= & {} \prod _{k=j}^{j} a_k Q \left[ \lambda _j,\lambda _i \right] ^{0} \\&+\prod _{k=j}^{i-1} a_k \sum _{s=1}^{0} \mathop {\sum _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{s}\\= & {} a_j. \end{aligned}$$
For the polynomial P we require that q be an integer or that \(i-j\) to be even: \(1-q \ge 0,\) \(1- \frac{i-j}{2} \ge 0,\) \(2-i+j \ge 0,\) \(2+j \ge i>j,\) \(i=j+2,\) \(q=1.\) Substituting these values in (14)
$$\begin{aligned} c^1_{j+2,j}= & {} \prod _{k=j}^{j+1} a_k Q \left[ \lambda _j,\lambda _i \right] ^{-1} + \prod _{k=0}^{0} b_{j+2k} P \left[ \lambda _j,\lambda _i \right] ^{0} +\\&+ \prod _{k=j}^{i-1} a_k \sum _{s=1}^{\lceil 0 \rceil } \mathop {\sum _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{-1+s},\\= & {} b_j. \end{aligned}$$
For R we want that: \(1-i+j+s \ge 0,\) \(\frac{1+s}{2}\ge \frac{i-j}{2}=q,\) \(\frac{1+s}{2}-1\ge q-1,\) \(\frac{s}{2}\ge \frac{s-1}{2}\ge q-1.\) From the limits of the first sum of the third term we have that \(\lceil q-1\rceil \ge s\ge 1\), \(\left\lceil \frac{s}{2}\right\rceil \ge \lceil q-1\rceil \ge s.\) Then, the only possible case is \(s=1\), then from the previous expression \(q=2\) and \(q=1\) from the second inequality, which is a contradiction. Therefore, for \(n=1\) there is not polynomial R or product of a’s times b’s. Thus, the formula (14) is valid for \(n=1\).
Let us assume as induction hypothesis that such formula is valid for \(n=K\), that is,
$$\begin{aligned}&c^K_{i,j} = \prod _{k=j}^{i-1} a_k Q\left[ \lambda _j,\lambda _i \right] ^{K-i+j}+ \prod _{k=0}^{q-1} b_{j+2k} P \left[ \lambda _j,\lambda _i \right] ^{K-q} \nonumber \\&\quad +\prod _{k=j}^{i-1} a_k \sum _{s=1}^{\lceil q-1 \rceil } \mathop {\sum _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+j+s} \end{aligned}$$
(15)
Let us check that it holds true for \(n=K+1\). To do that, one has to analyze the three cases shown in Fig. 1. Due to the similarities in the proofs we only present the proof for the first case and when \(\frac{i-j}{2}\) is an integer.
Fig. 1

possible scenarios for \(A^{K+1}=A A^K\)

Let us assume that q is an integer. Since q depends on the values of i and j we define three values of q for each one of the three entries that are multiplying in matrix \(A^K\),
$$\begin{aligned} q=\frac{i-j}{2} \text {,} q^+=\frac{i-1-j}{2} \text {,} q^-=\frac{i-2-j}{2}=q-1, \end{aligned}$$
Moreover, \(c_{i,i}=\lambda _i\), \(c_{i,i-1}=a_{i-1}\), \(c_{i,i-2}=b_{i-2}\). Multiplying A by \(A^K\) we get
$$\begin{aligned} c^{K+1}_{i,j}= & {} \sum _{m=1}^{\infty } c_{i,m} c^{K}_{m,j} = c_{i,i-2} c^K_{i-2,j}+c_{i,i-1} c^K_{i-1,j}+c_{i,i} c^K_{i,j} \\= & {} b_{i-2} c^K_{i-2,j}+a_{i-1} c^K_{i-1,j}+\lambda _i c^K_{i,j}. \end{aligned}$$
Using the induction hypothesis and the notation of q, \(q^+\) y \(q^-\) we obtain
$$\begin{aligned}&c^{K+1}_{i,j}=b_{i-2} \prod _{k=j}^{i-3} a_k Q \left[ \lambda _j,\lambda _{i-2} \right] ^{K-i+2+j} + b_{i-2} \prod _{k=0}^{q^- -1} b_{j+2k} P \left[ \lambda _j,\lambda _{i-2} \right] ^{K-q^-} \\&\qquad +\, b_{i-2} \prod _{k=j}^{i-3} a_k \sum _{s=1}^{\lceil q^- -1 \rceil } \mathop {\sum _{i_r=i_{r-1}+2}^{i-4}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _{i-2},\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+2+j+s}\\&\qquad +\,a_{i-1} \prod _{k=j}^{i-2} a_k Q \left[ \lambda _j,\lambda _{i-1} \right] ^{K-i+1+j} + a_{i-1} \prod _{k=0}^{q^+ -1} b_{j+2k} P \left[ \lambda _j,\lambda _{i-1} \right] ^{K-q^+}\\&\qquad +\,a_{i-1} \prod _{k=j}^{i-2} a_k \sum _{s=1}^{\lceil q^+ -1 \rceil } \mathop {\sum _{i_r=i_{r-1}+2}^{i-3}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+1+j+s}\\&\qquad + \,\lambda _i \prod _{k=j}^{i-1} a_k Q \left[ \lambda _j,\lambda _{i} \right] ^{K-i+j} +\lambda _i \prod _{k=0}^{q-1} b_{j+2k} P \left[ \lambda _j,\lambda _i \right] ^{K-q}\\&\qquad + \,\lambda _i \prod _{k=j}^{i-1} a_k \sum _{s=1}^{\lceil q-1 \rceil } \mathop {\sum _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+j+s}. \end{aligned}$$
Since q is an integer, then \(q^+\) is not an integer and \(\lceil q^+ -1 \rceil =\lceil q-1 \rceil \) and \(\lceil q^- -1 \rceil =\lceil q-2 \rceil . \) Therefore, the fifth sum is zero and
$$\begin{aligned} c^{K+1}_{i,j}= & {} b_{i-2} \prod _{k=j}^{i-3} a_k Q \left[ \lambda _j,\lambda _{i-2} \right] ^{K-i+2+j} + b_{i-2} \prod _{k=0}^{q-2} b_{j+2k} P \left[ \lambda _j,\lambda _{i-2} \right] ^{K-q+1}\\&+ \, b_{i-2} \prod _{k=j}^{i-3} a_k \sum _{s=1}^{\lceil q-2 \rceil } \mathop {\sum _{i_r=i_{r-1}+2}^{i-4}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _{i-2},\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+2+j+s}\\&+ \prod _{k=j}^{i-1} a_k Q \left[ \lambda _j,\lambda _{i-1} \right] ^{K-i+1+j}\\&+ \prod _{k=j}^{i-1} a_k \sum _{s=1}^{\lceil q-1 \rceil } \mathop {\sum _{i_r=i_{r-1}+2}^{i-3}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+1+j+s}\\&+ \,\lambda _i \prod _{k=j}^{i-1} a_k Q \left[ \lambda _j,\lambda _i \right] ^{K-i+j} +\lambda _i \prod _{k=0}^{q-1} b_{j+2k} P \left[ \lambda _j,\lambda _i \right] ^{K-q}+\\&+ \,\lambda _i \prod _{k=j}^{i-1} a_k \sum _{s=1}^{\lceil q-1 \rceil } \mathop {\sum _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+j+s}. \end{aligned}$$
To simplify, observe that if \(i-2=j+2k\), then \(k=q-1\). “grouping“ the fourth sum with the sixth and the seventh with the second
$$\begin{aligned} c^{K+1}_{i,j}= & {} \prod _{k=j}^{i-1} a_k \left( Q \left[ \lambda _j,\lambda _{i-1} \right] ^{K-i+1+j} +\lambda _i Q \left[ \lambda _j,\lambda _{i} \right] ^{K-i+j} \right) \\&+ \prod _{k=0}^{q-1} b_{j+2k} \left( \lambda _i P \left[ \lambda _j,\lambda _i \right] ^{K-q} + P \left[ \lambda _j,\lambda _{i-2} \right] ^{K-q+1} \right) \\&+ \frac{b_{i-2}}{a_{i-2} a_{i-1}} \prod _{k=j}^{i-1} a_k \sum _{s=1}^{\lceil q-2 \rceil } \mathop {\sum _{i_r=i_{r-1}+2}^{i-4}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _{i-2},\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+2+j+s}\\&+ \prod _{k=j}^{i-1} a_k \sum _{s=1}^{\lceil q-1 \rceil } \mathop {\sum _{i_r=i_{r-1}+2}^{i-3}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+1+j+s}\\&+ \frac{b_{i-2}}{a_{i-2} a_{i-1}} \prod _{k=j}^{i-1} a_k Q \left[ \lambda _j,\lambda _{i-2} \right] ^{K-i+2+j}\\&+ \,\lambda _i \prod _{k=j}^{i-1} a_k \sum _{s=1}^{\lceil q-1 \rceil } \mathop {\sum _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+j+s} \end{aligned}$$
Reordering and using properties of the polynomials P, Eq. (12), and Q, Eq. (11), writing \(Q \left[ \lambda _j,\lambda _{i-2} \right] ^{K-i+2+j}= R \left[ \lambda _j,\lambda _{i-1},\lambda _{i-1} \right] ^{K-i+2+j}\) and factorizing the product of a’s we get:
$$\begin{aligned} c^{K+1}_{i,j}= & {} \prod _{k=j}^{i-1} a_k Q \left[ \lambda _j,\lambda _i \right] ^{K-i+1+j}+ \prod _{k=0}^{q-1} b_{j+2k} P \left[ \lambda _j,\lambda _i \right] ^{K-q+1}\\&+ \prod _{k=j}^{i-1} a_k \Bigg \{ \frac{b_{i-2}}{a_{i-2} a_{i-1}} R \left[ \lambda _j,\lambda _{i-1},\lambda _{i-1} \right] ^{K-i+2+j} \\&+ \frac{b_{i-2}}{a_{i-2} a_{i-1}} \sum _{s=1}^{q-2} \mathop {\sum _{i_r=i_{r-1}+2}^{i-4}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _{i-2},\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+2+j+s}\\&+ \sum _{s=1}^{q-1} \mathop {\sum _{i_r=i_{r-1}+2}^{i-3}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+1+j+s} \\&+ \sum _{s=1}^{q-1} \mathop {\sum _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} \lambda _i R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+j+s} \Bigg \} \end{aligned}$$
Next, the first three terms inside the brackets are simplified. Some details are omitted in order to shorten the proof.
$$\begin{aligned} c^{K+1}_{i,j}= & {} \prod _{k=j}^{i-1} a_k Q \left[ \lambda _j,\lambda _i \right] ^{K-i+1+j}+ \prod _{k=0}^{q-1} b_{j+2k} P \left[ \lambda _j,\lambda _i \right] ^{K-q+1} + \prod _{k=j}^{i-1} a_k \\&\times \Bigg \lbrace \sum _{s=1}^{q-1} \mathop {\sum _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+1+j+s} \\&+ \sum _{s=1}^{q-1} \mathop {\sum _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} \lambda _i R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+j+s} \Bigg \rbrace . \end{aligned}$$
Grouping the second and third terms we have
$$\begin{aligned} c^{K+1}_{i,j}= & {} \prod _{k=j}^{i-1} a_k Q \left[ \lambda _j,\lambda _i \right] ^{K-i+1+j}+ \prod _{k=0}^{q-1} b_{j+2k} P \left[ \lambda _j,\lambda _i \right] ^{K-q+1} \\&+ \prod _{k=j}^{i-1} a_k \ \sum _{s=1}^{q-1} \mathop {\sum _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} \\&\times \bigg \lbrace R \left[ \lambda _j,\lambda _{i-1},\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+1+j+s}\\&+ \,\lambda _i R \left[ \lambda _j,\lambda _i,\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K-i+j+s} \bigg \rbrace \end{aligned}$$
Using the propertiess of polynomial R, Eq. (13), and since \(q-1=\lceil q-1 \rceil \),
$$\begin{aligned} c^{K+1}_{i,j}= & {} \prod _{k=j}^{i-1} a_k Q \left[ \lambda _j,\lambda _i \right] ^{K+1-i+j}+ \prod _{k=0}^{\lceil q-1 \rceil } b_{j+2k} P \left[ \lambda _j,\lambda _i \right] ^{K+1-q} \\&+ \prod _{k=j}^{i-1} a_k \ \sum _{s=1}^{\lceil q-1 \rceil } \mathop {\sum _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}}{a_{i_k}a_{i_k+1}} R \left[ \lambda _j,\lambda _{i},\left\{ \lambda _{i_1+1},\ldots , \lambda _{i_s+1}\right\} \right] ^{K+1-i+j+s}, \end{aligned}$$
and we conclude that the formula (15) holds true for \(n=K+1.\) \(\square \)

The formula that we have obtained seems quite complex in general, but that is not the case in many situations. Moreover, it aims to greatly improve the accuracy of previous approximations for partial differential equations and to extend their range of applicability. It also allows us to directly obtain the value of the iterates of the DAR operator by evaluating such formula.

In order to establish a better illustration of the previous result we try a special case where the eigenvalues of the matrix A are different; the purpose of this is to offer numerical visualizations of the analytic formula and its relationship with other areas of approximation theory. It is also important to remark that this case covers a large set of problems appearing in applications.

Case with Different Eigenvalues

Assume that all the eigenvalues of the matrix A are different, then it can be diagonalized and its n-th power can be obtained using the following theorem.

Theorem 5.1

Assume that all the eigenvalues of the matrix A are different. Let us denote by \(\left( \Phi \right) _{i,j}\) the entry ij of \(\Phi \) defined as
$$\begin{aligned} \left\{ \begin{array}{ll} 0 &{} { if} \quad i<j, \\ 1 &{} { if} \quad i=j, \\ \prod \limits _{k=j}^{i-1} a_k \prod \limits _{r=j+1}^i \frac{1}{\left( \lambda _j - \lambda _r \right) }+\frac{b_j}{\left( \lambda _j-\lambda _{i}\right) }\prod \limits _{k=1}^{q-1} \frac{b_{j+2k}}{ \left( \lambda _j-\lambda _{j+2k}\right) }+ &{} { if} \quad i>j \quad and \quad q \in \mathbb {N},\\ \prod \limits _{k=j}^{i-1} a_k \prod \limits _{r=j+1}^i \frac{1}{\left( \lambda _j - \lambda _r \right) } \sum \limits _{s=1}^{ \lfloor q-1 \rfloor } \mathop {\sum \limits _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \prod \limits _{k=1}^s \frac{b_{i_k}\left( \lambda _j-\lambda _{i_k+1}\right) }{a_{i_k}a_{i_k+1}}, &{} \end{array}\right. \end{aligned}$$
where \(q=\frac{i-j}{2}\) if \(q \not \in \mathbb {N}\) then the second term vanishes. Let \(\Omega _{\omega }\) be a diagonal infinite matrix with \(\left( \Omega _{\omega } \right) _{ii}=\omega \left( \lambda _i \right) \), where \(\omega : \mathbb {R} \rightarrow \mathbb {R}\) is an arbitrary fixed function. Then \(\Phi \) is invertible (an inferior triangular infinite matrix) with entries \(\Phi ^{-1}_{i,j}\) given by
$$\begin{aligned} \left\{ \begin{array}{ll} 0 &{} { if}h \quad i<j, \\ 1 &{} { if} \quad i=j, \\ \prod \limits _{k=j}^{i-1} a_k \prod \limits _{r=j}^{i-1} \frac{1}{\left( \lambda _i-\lambda _r\right) } + \prod \limits _{k=0}^{q-1} \frac{b_{j+2k}}{\left( \lambda _i-\lambda _{j+2k}\right) } &{} \text{ if } \quad i>j, q \in \mathbb {Z}, \\ \prod \limits _{k=j}^{i-1} a_k \prod _{r=j}^{i-1} \frac{1}{\left( \lambda _i-\lambda _r\right) } \sum \limits _{s=1}^{\lfloor q-1 \rfloor } \mathop {\sum \limits _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \prod _{k=1}^s \frac{b_{i_k}\left( \lambda _i-\lambda _{i_k+1}\right) }{a_{i_k}a_{i_k+1}} &{} \end{array}\right. \end{aligned}$$
Fig. 2

a Non zero convergence. b Convergence to zero

where again if \(q \not \in \mathbb {N}\) then the second term vanishes. Moreover, \(A^n= \Psi \cdot \Omega _{\omega } \cdot \Phi \) with the choice \(w(x)=x^n\) for \(n=0,1,2,3,...\) and its entries are given by
$$\begin{aligned} \begin{array}{ll} 0 &{} { if} \quad i<j \\ \omega (\lambda _{i}) &{} { if} \quad i=j \\ \left( \frac{1}{q} \prod \limits _{k=0}^{q-1} b_{j+2k} \right) \frac{d^q}{d \lambda ^q}\mathcal{L}_\omega [\lambda _{j},\lambda _{j+2},\dots ,\lambda _{i}] + &{} { if} \quad i>j\\ \left( \frac{1}{2q} \prod _{k=j}^{i-1} a_{k} \right) \frac{d^{2q}}{d \lambda ^{2q}}\mathcal{L}_\omega [\lambda _{j},\lambda _{j+1},\dots ,\lambda _{i-1},\lambda _{i}]+ &{} and \quad q \in \mathbb {Z} \\ \sum \limits _{s=1}^{\lfloor q-1 \rfloor } \left( \frac{1}{q+s} \mathop {\sum \limits _{i_r=i_{r-1}+2}^{i-2}}_{r=1,s} \prod \limits _{k=1}^s b_{i_k} \mathop {\prod \limits _{\ell =j}^{i-1} a_{\ell }}_{\ell \ne a_{i_k},a_{i_k+1}} \right) \frac{d^{q+s}}{d \lambda ^{q+s}}\mathcal{L}_\omega \left[ \lambda _{j},\lambda _{i},\left\{ \lambda _{i_1+1},\ldots ,\lambda _{i_s+1} \right\} \right] \end{array} \end{aligned}$$
where
  • \(\mathcal{{L}}_\omega [\lambda _j,\lambda _{j+2},\ldots , \lambda _i]\) is the Lagrange polynomial on \(\lambda \) of degree q for the nodes \(\{ \lambda _j,\lambda _{j+2},\ldots , \lambda _i \}\) for the function \(\omega ,\) [3].

  • \(\mathcal{{L}}_\omega \left[ \lambda _j,\lambda _{j+1},\ldots , \lambda _{i-1},\lambda _i \right] \) is the Lagrange polynomial on \(\lambda \) of degree 2q for the nodes \(\{ \lambda _j,\lambda _{j+2},\ldots ,\lambda _{i-1},\lambda _i \}\) for the function \(\omega ,\).

  • \(\mathcal{L}_\omega [\lambda _{j},\lambda _{i},\left\{ \lambda _{i_1+1},\ldots ,\lambda _{i_s+1} \right\} ]\) is the Lagrange polynomial on \(\lambda \) of degree \(q+s\) for the nodes \(\{ \lambda _j,\lambda _{j+1},\ldots ,\lambda _{i-1},\lambda _i \}\) except for the nodes \(\lambda _{i_1+1},\ldots ,\lambda _{i_s+1}\) for the function \(\omega \).

For the sake of completeness now we incorporate two numerical examples that illustrate the behavior of a simple sequence under the action of the DAR operator. We start by choosing \({\mathbf \lambda }\) given by \( \left\{ \frac{1}{n^2} \right\} _{n=2}^\infty \) and the sequences \({\mathbf a}\) and \({\mathbf b}\) given by constant sequences with values 0.595 and 0.5 respectively. The initial condition is given by \( {\mathbf s}=(1,0,0, \dots ),\) which can be considered as an initial disturbance for the reaction-diffusion-advection equation. In Fig. 2a) we show several stages of the behavior of the orbit of \({\mathbf s},\) by applying one, two, three, four, ten and thirty iterations of the DAR operator to the sequence \({\mathbf s}.\) We can observe how the initial disturbance, \({\mathbf s}, \) is shifted to the right and its amplitude increases. We can modify the first example by choosing again \({\mathbf \lambda }\) and \({\mathbf b}\) as before. \({\mathbf a}\) is chosen as \( \{ \frac{1}{n^3} \}_{n=2}^\infty \). In this case, the iteration sequence converges to a zero sequence. In Fig. 2b) we present the first four iterations of the DAR operator. Notice how the initial disturbance, which has only a nonzero component, transforms into a convergent sequence where all its components converge to a zero value.

Footnotes

  1. 1.

    A polynomial with all its coefficients equal to one.

References

  1. 1.
    Asheghi, R.: Bifurcations and dynamics of a discrete predator-prey system. J. Biol. Dyn. 8(1), 161–186 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bischi, G.I., Naimzada A.K.: A Kaleckian macromodel with memory. In: Cristini, A., Fazzari, S., Greenberg, L., Leoni R. (eds) Cycles, Growth and the Great Recession, Routledge, United Kingdom, pp 103–116 (2015)Google Scholar
  3. 3.
    Burden, R.L.: Numerical analysis. Beooks/Cole, Australia (2001)Google Scholar
  4. 4.
    Cortés, J.C., Navarro-Quiles, A., Romero, J.V., Roselló, M.D., Sohaly, M.A.: Solving the random Cauchy one-dimensional advection-diffusion equation: numerical analysis and computing. J. Comput. Appl. Math. 330, 920–936 (2018).  https://doi.org/10.1016/j.cam.2017.02.001 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cortés, J.C., Jódar, L., Villafuerte, L., Villanueva, R.J.: Computing mean square approximations of random diffusion models with source term. Math. Comput. Simul. 76(1–3), 44–48 (2007).  https://doi.org/10.1016/j.matcom.2007.01.020. 12MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chong, T.H.: A variable mesh finite difference method for solving a class of parabolic differential equations in one space variable. SIAM J. Numer. Anal. 15, 835–857 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    El-Sayed, A.M.A., Elsadany, A.A., Awad, A.M.: Chaotic dynamics and synchronization of cournot duopoly game with a logarithmic demand function. Appl. Math. 9(6), 3083–3094 (2015)Google Scholar
  8. 8.
    Forsythe, G.E., Wasow, W.R.: Finite difference methods for partial differential equations. Wiley, New York (1964)zbMATHGoogle Scholar
  9. 9.
    Gladwell, I., Wait, R.: A survey of numerical methods for partial differential equations. Oxford University Press, New York (1979)zbMATHGoogle Scholar
  10. 10.
    Jerez, S., Gonzalez, L.M., Solis, F.J.: A regular perturbation analytical-numerical method for the evolution of precancerous lesions caused by the human papillomavirus. Numer. Methods Partial Differ. Eq. 31(3), 847–855 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Khodabin, M., Maleknejad, K., Rostami, M., Nouri, M.: Numerical solution of stochastic differential equations by second order Runge-Kutta methods. Math. Comput. Model. 53, 1910–1920 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li, B., He, Z.: Bifurcations and chaos in a two-dimensional discrete HindmarshRose model. Nonlinear Dyn. 76(1), 697–715 (2014)CrossRefzbMATHGoogle Scholar
  13. 13.
    Maire, S., Nguyen, G.: Stochastic finite differences for elliptic diffusion equations in stratified domains. Math. Comput. Simul. 121, 146–165 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pareek, N.K., Patidar, V., Sud, K.K.: Discrete chaotic cryptography using external key. Phys. Lett. A 309, 75–82 (2003).  https://doi.org/10.1016/S0375-9601(03)00122-1. 12, 17MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Reynolds Jr., A.C.: Convergent finite difference schemes for non-linear parabolic equations. SIAM J. Numer. Anal. 9, 523–533 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Schulman, L.S., A Seiden, P.E.: Statistical mechanics of a dynamical system based on Conways game of Life. J. Stat. Phys. 19(3), 293–314 (1978)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Solis, F., Gonzalez, L.M.: A model for HVP infected cells at different lesion discrete stages. Int. J. Complex Syst. Sci. 2(1), 7–10 (2012)Google Scholar
  18. 18.
    Solis, F., Gonzalez, L.M.: Modeling the effects of Human Papilloma Virus in cervical cells. Int. J. Comput. Math. 91, 1–9 (2013)Google Scholar
  19. 19.
    Solis, F., Barradas, I.: Discrete multiple delay advection-reaction operators. J. Comput. Appl. Math. 291, 441–448 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Solis, F.: Dynamical properties of families of discrete delay advection-reaction operators. J. Differ. Equ. Appl. 19(8), 1218–1226 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Solis, F.: Families of discrete advection-reaction operators via divided differences. Appl. Math. Lett. 25, 775–778 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Vasilyev, R.V., et al.: Solution of the stokes equation in three-dimensional geometry by the finite-difference method. Math. Models Comput. Simul. 8(1), 63–72 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Weiser, A., Wheeler, M.F.: On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25(2), 351–375 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2018

Authors and Affiliations

  1. 1.CIMATGuanajuato GtoMexico

Personalised recommendations