Abstract
Chebyshev’s and Legendre’s differential equations’ solutions are solved employing the differential transform method (DTM) and the power series method (PSM) in this study. This research shows that this method is efficient and effective in discovering Chebyshev’s and Legendre’s differential equation (DE) series solutions and that it can reduce calculation size when compared to other methods.
1. Introduction
Pukhov [1] first proposed the differential transform method (DTM) in 1982. In the meantime, Zhou [2] introduced the DTM theory in 1986, which was utilized to solve nonlinear and linear initial value problems (IVP) in electric circuit analysis. The Taylor series [2] was used to develop the DTM, which was touted as a new method. It resembles a semianalytical technique that employs the Taylor series to generate power series representations of differential equation solutions. The method is effective for obtaining approximate and exact solutions to a linear [3] and nonlinear [4] ordinary differential equations’ (ODEs) system. It is an iterative procedure for gaining differential equations’ analytic series solutions. In several previous works, the DTM was created to solve many types of integral and differential equations. Ali [5] created the DTM to solve partial differential equations (PDE), whereas Ayaz [6, 7] implemented it to differential algebraic equations.
Arikoglu and Ozkol [8] also used the DTM to solve integrodifferential equations with boundary values’ conditions (BVCs). Furthermore, Odibat et al. [9] employed the DTM to solve separable kernel Volterra integral equations. The DTM was used by Tari and Ziyace [3] in solving a system of 2D nonlinear Volterra integrodifferential equations. Here, the DTM was used to solve systems of integral and integro-differential equations, fractional differential equations, multi-order fractional differential equations, time-fractional diffusion equation and the singularly perturbed Volterra integral equations [5, 9–15]. Nonlinear parabolic-hyperbolic PDEs have also been studied using the DTM. In works by Biazar and Abdul Halim-Haasan [10] and also Bervillier [11] solved the two-dimensional Fredholm integral equations. Meanwhile, Abdewahaid [16] proposed a new 1D differential transform basic formula. Meanwhile, based on the DTM [17], El-moneam, Badr, and Ahmed Msmali proposed a general scheme to solve linear first-order DE systems. Some types of Euler–Cauchy ODE have been solved using DTM by El-moneam et al. [18]. Moreover, Esmail Hesameddini and Amir Peyrovi solved Chebyshev and Legendre equations using the homotopy perturbation method (HPM) [19].
For two special nonconstant coefficient ODES, namely, the Chebyshev and Legendre equations, this study uses the PSM and DTM.
Chebyshev and Legendre polynomials have several applications in physics and mathematics. Chebyshev polynomials are essential in approximation theory, particularly, in a polynomial interpolation general theory that yields an approximate polynomial for continuous functions having a minimum error norm. Note that Chebyshev polynomials are obtained by solving the Chebyshev DE:
The equation above is known as the first kind of polynomials denoted as ; meanwhile,resembles the polynomial of the second kind expressed as , in which n refers to a constant. The usual PSM can be used to solve Chebyshev’s DE. At , these equations exhibit singular points. As a result, the series solution of equations converges only around the origin for . denotes a polynomial in x of degree n expressed by the relation, where n denotes a positive integer of Chebyshev’s polynomial:
Conversely, the second kind polynomial is a degree n polynomial in x defined by
The first- and second-kind polynomials are the result of certain simple calculations:
Furthermore, via (3) and (4), the equations given below for as well as are obtained as below:
Refaat and Mason and Handscomb [20, 21] include information on relations between first and second kinds of polynomials, Chebyshev polynomial zeros, integrals, orthogonality, and derivatives, as well as many other features.
The Legendre DE is the other equation that will be explored in this work,in which n represents a constant. Legendre polynomials are those that have a nonnegative integer solution to these equations. The PSM, such as Chebyshev, can be used to find equations over a range of values (−1, 1). The Newtonian potential theory relies heavily on the solutions to these equations. Legendre polynomials also appear in PDEs when the Laplace equation .
Legendre polynomials are defined in another way by Rodrigues’ formula:
The Legendre polynomials can be obtained immediately from this definition:
Legendre’s function of fractional order, recursive definition, orthogonality, and other Legendre polynomial features can be found in [20, 21].
The DTM’s fundamental concepts will be explained in Section 2. Next, in Section 3, we solve Legendre’s equation using the DTM and compare the result to the solution obtained using PSM in Section 4. Finally, in Section 5, we use the DTM to solve Chebyshev’s equation and compare it to the solution obtained using the PSM in Section 6.
2. Analysis of the Differential Transform Method (DTM)
In [1, 2, 22, 23], the DTM’s fundamental concept and theorems, as well as its applicability to numerous types of DE, are given. We shall give a review of the DTM for the reader’s convenience. In doing so, we assume that function in which denotes any point in interval I. Here, Taylor’s series of about can, therefore, be represented by
Definition 1. If is an analytic function on , then ’s order differential transform is expressed asNotice that, ’s inverse differential transform is expressed as
Theorem 1. Let and refer to analytic functions having differential transforms and accordingly; thus,in which and represent constants.
Theorem 2. Let denote an analytic function, with differential transform ; then,
Theorem 3. Let ; then, , where is the Kronecker delta.
Theorem 4. Let and express analytic functions in which
Thus, the function ’s differential transform is expressed as
Theorem 5. Let as well as represent analytic functions, in which ; then,The proves of the following results can be found in [4].
Theorem 6. Let resembles an analytic function with thus,and if , then
Theorem 7. Let represent an analytic function, in which then,and
Theorem 8. Let become an analytic function, where then,Notice that, for , (21) reduces to the formula given below:
Theorem 9. Let be an analytic function, while then,
3. The Solutions of Legendre’s Differential Equation
Now, consider the initial value problem given byreferred to as the Legendre differential equation, in which and are constants, whereas n refers to an integer.
We use the differential transform to solve the IVP (24) and (25):
Next, using the differential transform properties (14) and (17) enables us to find
This leads to the recurrence relation:
The recurrence relation (28) givesfor The recurrence relation is, therefore,
Let j take on the values then, the recurrence relation (30) yieldsand so on. If we choose Y(0) = 1 and Y(1) = 0, we obtain the solution
Meanwhile, choosing Y(0) = 0 and Y(1) = 1, we determine a second solution given by
Because both series converge for , their convergence radius is unity. Since (32) contains only even power of while (33) contains only odd powers, (24) general solution may be expressed as follows:
Note that if refers to an even integer, then the first series vanishes; meanwhile, denotes an infinite series. For instance, if , hence
Likewise, the series for vanishes with when n is an odd integer. To put it in another way, we get an -degree polynomial solution of the Legendre equation when n is a nonnegative integer.
A solution is a constant multiple of a Legendre equation solution. Note that it is advisable to use specific values for or , which depends on whether n denotes an odd or even positive integer.
For , we select ; meanwhile, for whereas, for , we choose , and for
For example, when , we have
Legendre polynomials, written as , are specified -degree polynomial solutions.
We may deduce that the first several Legendre polynomials are represented by the series for and and from the above choices of and as shown below:
We can demonstrate that these polynomials are particular solutions of the DE:
4. The Solution of Legendre’s Equation Based on the Power Series Method
We solve the Legendre (24) with the PSM and compare our results to the power series results.
There exist two linearly independent solutions of the form since resembles an ordinary point of (24). We discover that when we substitute
Upon collecting terms gives
The recurrence relation is, therefore, given by
If we choose a and for , we get the solution:
Alternatively, we can find a second solution by setting and :
Since (44) contains only the even power of and (45) has only odd powers, equation (24) general solution can be expressed as
Solutions (44) and (45) are the same results obtained in equations (32) and (33) if the DTM is applied.
5. The Solution of Chebyshev’s Differential Equation (DE)
Chebyshev’s DE is given by
The Chebyshev DE possess regular singular points at -1, 1, as well as in the case of . By employing the differential transform to both sides of (47), it can be solved utilizing the DTM:
Then,
From this, we obtain, for the even coefficients,
And for the odd coefficients,
The coefficients that are even expressed as
And for odd coefficients given as
The general solution is then given by summing over all indices:
This may be accomplished in the closed form as follows:
Changing the variables yields the solution’s equivalent form:where refers to a Chebyshev polynomial of the second kind whereas refers to a Chebyshev polynomial of the first kind.
The solution can also be written in another way:
Another way to solve (47) is by changing variables that give the equivalent forms of (47) as follows.
By letting , we get the following differential equation:
Meanwhile, for , (43) can be reduced to the following differential equation given by
When both sides of (58) and (59) are applied with the differential transform, the result yieldsand
Let be the recurrence relation (60); it then gives
Then, the solution of (44) is given byor
Equivalently,in which and denote Chebyshev polynomials of the first and second kinds, correspondingly, of degree n.
Similarly, from (61), we obtainor
Equivalently,
6. The Solution of Chebyshev’s Equation by Using the Power Series Method (PSM)
Consider Chebyshev’s equation given by
Regular singular points in the Chebyshev differential equation for are -1, 1, and . The PSM can solve it via the expansions:
Now, we plug (70)–(72) into (69) to obtain
So,and by induction,
Since (74) and (75) are special cases of (76), the general recurrence relation can be written as
From this, we obtain, for the even coefficients,and for the odd coefficients,
The even coefficients can be written as
And for odd coefficients given as
The general solution is then given by summing over all indices:
This may be accomplished in the closed form as follows:
Changing the variables yields the solution’s equivalent form.where denotes Chebyshev’s polynomial of the second kind, whereas refers to Chebyshev polynomial of the first kind.
The findings produced using the DTM are identical.
7. Conclusion
We successfully used the DTM to solve Chebyshev’s and Legendre’s DE in this study. The current method eliminates the computing challenges associated with previous traditional methods by allowing all computations to be computed using simple manipulations. As a result, this method can be used to solve a variety of DE without the need for discretization, linearization, or perturbation. Also, the solution of Chebyshev’s equation is obtained using the power series method. In comparison between these two methods, the results are identical to the results obtained via the differential transform method.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.