1. Introduction
The heat equation pioneered by Fourier [
1] describes the distribution of heat in a given body over time [
2], which is a type of second-order parabolic partial differential equation. It has many applications in diverse scientific fields. Moreover, it has been studied analytically and numerically. For example, Meyu and Koriche [
3] proposed two techniques based on the separation of variables and finite-difference methods to solve the heat equation in one dimension. Liu and Chang [
4] used a method of nonlocal boundary shape functions to solve a nonlinear heat equation with nonlocal boundary conditions. Tassaddiq et al. [
5] introduced an approximate approach based on a cubic B-spline collocation method to solve the heat equation with classical and nonclassical boundary conditions.
It is well-known that obtaining accurate and efficient methods for solving differential equations has become an important research point. There are several analytic and numerical methods, such as the homotopy analysis method [
6,
7], the variational iteration method [
8,
9], the Adomian decomposition method [
10,
11], the finite-difference method [
12,
13,
14], the finite-element method [
15,
16,
17], and spectral methods [
18,
19,
20,
21,
22]. Spectral methods have many advantages if compared with the other methods because they yield exponential rate convergence, a good accuracy, and the computational efficiency of the solutions while failing for many complicated problems with singular solutions. Thus, it is relevant to be interested in how to enlarge the adaptability of spectral methods and construct certain simple approximation schemes without a loss of accuracy for more complicated problems. Further applications of spectral methods in different disciplines may be found in [
23,
24,
25,
26,
27,
28,
29].
Orthogonal polynomials, such as Legendre polynomials and Chebyshev polynomials, have received a lot of attention from both theoretical and practical perspectives [
30,
31,
32]. Chebyshev polynomials have been used as an important category of basis functions to solve ordinary, partial, and fractional differential equations, see for instance [
33,
34,
35,
36,
37,
38]. Two major reasons for the widespread use of these polynomials are the high accuracy of the approximation and the simplicity of numerical methods established based on these polynomials. There are six types of Chebyshev polynomials, they are Chebyshev polynomials of the first, second, third, fourth, fifth, and sixth kind. All the kinds of Chebyshev polynomials have their important parts in numerical analysis and approximation theory. There are old and recent contributions regarding the first four kinds, see, for example, [
39,
40], while the fifth and sixth kind of Chebyshev polynomials have gained recently a fast-growing attention from many authors. For instance, Sadri and Aminikhah in [
41] treated a multiterm variable-order time-fractional diffusion-wave equation using a new efficient algorithm based on the
. Moreover, Abd-Elhameed and Youssri in [
42] employed the
for solving the convection–diffusion equation.
The following items are the main goals of this paper:
Deriving new theorems, corollaries, and lemmas concerned with the shifted that serve in the derivation of our proposed numerical scheme.
Presenting a new spectral tau algorithm for the numerical treatment of the heat conduction equation.
Investigating the convergence analysis of the proposed double-shifted Chebyshev expansion.
Performing some comparisons to clarify the efficiency and accuracy of our method.
To the best of our knowledge, some advantages of the proposed technique can be mentioned as follows:
By choosing the shifted as basis functions, and taking a few terms of the retained modes, it is possible to produce approximations with excellent precision. Less calculation is required. In addition, the resulting errors are small.
In comparison to other Chebyshev polynomials, the shifted are not as well-studied or used. This motivates us to find theoretical findings concerning them. Furthermore, we found that the obtained numerical results, if they are used as basis functions, are satisfactory.
We point out here that the novelty of our contribution in this paper can be listed as follows:
Some derivatives and integral formulas of the shifted are given in reduced formulas that do not involve any hypergeometric forms.
The employment of these basis functions to the numerical treatment of the heat conduction equation is new.
The contents of the paper are arranged as follows.
Section 2 is devoted to presenting mathematical preliminaries containing some relevant properties of
and their shifted ones. In addition, some new formulas concerning the shifted
are derived. In
Section 3, we present and implement a spectral tau method for solving the heat conduction equation based on employing the shifted
. In
Section 4, we investigate in detail the convergence and error analysis of the suggested shifted
. In
Section 5, some numerical examples are given to ensure the efficiency, simplicity, and applicability of the suggested method. Finally, conclusions are reported in
Section 6.
2. An Account on the Shifted and Some New Useful Formulas
This section is confined to presenting an account on the , , , and their shifted ones. In addition, building on some of their fundamental relations, we derive some new specific formulas that serve in the derivation of our proposed numerical scheme. More precisely, we establish the second-order derivative formulas of the shifted polynomials and also the corresponding integral formulas of these polynomials.
2.1. An Account on the Shifted
The
are a sequence of orthogonal polynomials on
(see, [
28,
43]) that satisfy the following orthogonality relation:
where
and
may be generated with the aid of the following recursive formula:
where
and
The shifted orthogonal
on
are defined as
with the orthogonality relation
where
and
Lemma 1 ([
28]).
The analytic formula of may be split to the following two analytic formulas: Theorem 1 ([
28]).
The following two inversion formulas hold for the polynomials : 2.2. Derivation of the Second-Order Derivative Formulas of
The following theorem exhibits the expressions of the second-order derivatives of in terms of their original ones.
Theorem 2. The second-order derivative of the polynomials can be expressed explicitly as: Proof. First, we prove relation (
6). The power-form representation of
in (
2) enables one to express
in the following form:
which can be written with the aid of the inversion formula (
4) as
The last relation after expanding and rearranging the terms can be converted into
Now, in order to reduce the summation on the right-hand side of the last formula, set
The application of Zeilberger’s algorithm mentioned in [
44] enables us to get the following recurrence relation for
:
with the initial values:
The recurrence relation (
8) can be exactly solved to give
and therefore, relation (
6) can be obtained.
Now, we prove Formula (
7). Based on relation (
3), we have
Making use of Formula (
5) yields
The last relation after expanding and rearranging the terms can be converted into
Now, set
and utilize again Zeilberger’s algorithm to show that
satisfies the following recurrence relation:
with the initial values:
The recurrence relation (
9) can be exactly solved to give
and therefore, relation (
7) can be obtained. ◻
As a result of Theorem 2, the formula expressing the derivatives of the can be merged to give the following result.
Corollary 1. Let The second-order derivative of the polynomials can be expressed explicitly as:where Now, the second-order derivatives of the shifted polynomials can be easily deduced. The following corollary exhibits this result.
Corollary 2. Let The second-order derivative of the polynomials can be expressed explicitly as:where Proof. The result is a direct consequence of Corollary 1 by replacing t by . ◻
2.3. Derivation of Integral Formulas of
In this section, new integral formulas of are derived in detail. For this derivation, the following two lemmas are useful.
Lemma 1. Let and . One haswhere Lemma 2. Let and . One has Proof. The proofs of Lemmas 1 and 2 can be done through some algebraic manipulations along with Zeilberger’s algorithm [
44]. ◻
Theorem 3. For all the following integral formulas hold: Proof. We prove formula (
11). The power-form representation (
2) enables one to express
as
In virtue of relation (
5), the last equation may be written alternatively as
After rearranging and expanding the terms in the previous equation, one gets
Thanks to Lemmas 1 and 2, we get the desired relation (
11).
Relation (
12) can be similarly proved through some algebraic computations. ◻
The following corollary is a direct consequence of Theorem 3.
Corollary 3. For all the following integrals formulas holdwhere and are constants.