Highly Accurate Method for Boundary Value Problems with Robin Boundary Conditions

The main aim of the current paper is to construct a numerical algorithm for the numerical solutions of second-order linear and nonlinear differential equations subject to Robin boundary conditions. A basis function in terms of the shifted Chebyshev polynomials of the first kind that satisfy the homogeneous Robin boundary conditions is constructed. It has established operational matrices for derivatives of the constructed polynomials. The obtained solutions are spectral and are consequences of the application of collocation method. This method converts the problem governed by their boundary conditions into systems of linear or nonlinear algebraic equations, which can be solved by any convenient numerical solver. The theoretical convergence and error estimates are discussed. Finally, we support the presented theoretical study by presenting seven examples to ensure the accuracy, efficiency, and applicability of the constructed algorithm. The obtained numerical results are compared with the exact solutions and results from other methods. The method produces highly accurate agreement between the approximate and exact solutions, which are displayed in tables and figures.


Introduction
Boundary value problems (BVPs) are extremely important in describing many realistic problems with various applications.The most famous of these are the problems associated with Dirichlet, Neumann, and Robin boundary conditions.The latter type is considered one of the most difficult conditions facing researchers when dealing with this type of problem.Because of the difficulty of Robin's boundary conditions, research studies that discuss this type of BVP have not attracted much interest.The condition is named after the scientist Victor Gustav Robin, who was behind its origin [1].It is also referred to as the "third kind" of boundary conditions, and these conditions are a linear mixture of the solution and its derivative at the boundary points.The present paper focuses on the numerical approach to solving second-order BVP associated with Robin boundary conditions.This type of BVP is given as follows: subject to the Boundary Conditions where 1 , 2 , 1 , 2 , 1 and 2 are all constants.In the case of Robin type, all of these constants on the left side of conditions (1.2) and (1.3) are non-zero.While in the case of Dirichlet type, we have 1 = 2 = 0 , otherwise this BVP will be sub- ject to Neumann condition when 1 = 2 = 0 .Robin boundary conditions appear in many branches of applications, such as electromagnetic problems, where they are named impedance boundary conditions, and heat transfer problems, where they are named convective boundary conditions, as explained in [2].Such conditions play an essential role in the study of diffusion equation occurring in biology and chemistry field [3].
The three popular versions of spectral methods are the collocation, tau, and Galerkin methods.They have important roles in obtaining numerical solutions for various mathematical models.These methods provide very accurate approximate solutions to various kinds of differential equations with a relatively small number of unknowns.The choice of the most convenient version of these methods is based on the type of the investigated differential equation and also on the kind of boundary conditions governed by it.In these methods, the use of operational

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Journal of Nonlinear Mathematical Physics (2023) 30:1239-1263 matrices to build efficient algorithms to obtain accurate numerical solutions to various types of differential equations reduces the computational efforts.
The idea of an operational matrix of derivatives depends on the choice of convenient basis functions and expressing the first derivative of these in terms of their original ones (see for instance, [18][19][20][21][22][23][24]).In the Galerkin method, if the considered differential equation is subject to homogeneous initial and boundary conditions, the choice of basis functions must satisfy these conditions to ensure that the proposed approximate solution also satisfies these conditions.While in the case of nonhomogeneous conditions, the transformation process to the corresponding homogeneous form must be carried out first.In the collocation method, it is not necessary to choose the basis functions that satisfy the given conditions, but the best choice, as in the Galerkin method.In the tau method, the basis functions don't satisfy the given conditions.
Up to now, and to the best of our knowledge, a Galerkin operational matrix using any basis function that satisfies the homogeneous Robin boundary conditions is not known and is traceless in the literature.This partially motivates our interest in such an operational matrix.Another motivation is the utilizing of this type of operational matrix for the numerical treatment of BVP (1.1)- (1.3).The principal aims of this paper can be summarized as follows: (i) Constructing a new class of basis polynomials, named Robin-Modified Chebyshev polynomials, using generalized shifted Chebyshev polynomials of the first kind that satisfy the homogeneous Robin boundary conditions.(ii) Establishing operational matrices for derivatives of the constructed polynomials.(iii) Constructing numerical algorithm for solving BVP (1.1)-(1.3)based on employing collocation method together with the introduced operational matrices of derivatives.(iv) Estimating the error obtained for the approximate solution.
The paper is organized as follows.In Sect.2, the first-kind Chebyshev polynomials and their shifted ones are discussed.Section 3 is limited to constructing Robin-Modified Chebyshev polynomials of first-kind which satisfy the homogeneous Robin boundary conditions.Section 4 is limited to developing a new operational matrix of modified first-kind Chebyshev polynomials' derivatives to handle BVP (1.1)-(1.3).The use of collocation method to solve numerical approach for BVP (1.1)-(1.3) is examined in Sect. 5.The theoretical convergence and error estimates are discussed in Sect.6. Section 7 contains seven examples, as well as comparisons with several other methods from the literature.Finally, Sect.8 displays some conclusions.

An Overview on First-Kind Chebyshev Polynomials and Their Shifted Ones
The orthogonal Chebyshev polynomials of the first kind T n (x) have the following trigonometric definition (see, [25]) and they are satisfying the orthogonality relation These polynomials can be built by using the following recurrence relation with T 0 (t) = 1, T 1 (t) = t.The polynomials T n (t) are special ones of the Jacobi poly- nomials, P (,) Lemma 2.1 The power form representation of the modified Chebyshev polynomials can be represented as where ) Journal of Nonlinear Mathematical Physics (2023) 30:1239-1263 Proof The analytical form of known shifted Chebyshev polynomials of first kind T * n (x;0, 1) is given by The analytical expression of T * n (x;a, b) can be written in the form Substituting the relation to Eq.(2.7), expanding and collecting similar terms -and after some rather manipulation -we can deduce that T * (q) n (0;a, b), q ≤ n, has the form (2.5) and this completes the proof of Lemma 2.1.◻ As a direct consequence of Lemma 2.1, we get the known analytic form of the shifted Chebyshev polynomials: Note 2.1 Here, it is important to remember that the generalized hypergeometric function is defined as [26] where b j ≠ 0 , for all 1 ≤ j ≤ q.

Robin-Modified Chebyshev Polynomials of First-Kind
In this section, a novel kind of polynomials, it will symbol with k (x) , will be devel- oped, which we call "Robin-Modified Chebyshev polynomials of first-kind" in order to satisfy given form of Homogeneous Robin Boundary Conditions: (2.7)

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In this respect, we propose Robin-Modified Chebyshev polynomials of first-kind in the form where q k (x) has the form where A k and B k are unique constants such that k (x) satisfy the two conditions (3.1) and (3.2).Substitution of k (x) into (3.1) and (3.2) leads to the two linear equations in the two unknowns A k and B k : In particular, and for the special case, homogeneous Dirichlet conditions can be obtained by taking 1 = 2 = 1 and 1 = 2 = 0 .In such case, we have Also, homogeneous Neumann conditions can be considered as a special case of a Robin-type conditions (3.1) and (3.2), by taking 1 = 2 = 0, 1 = 2 = 1 , which immediately gives The proposed Robin-Modified Chebyshev polynomials of first-kind have the special values ) Journal of Nonlinear Mathematical Physics (2023) 30:1239-1263

Operational Matrix of Derivatives of Robin-Modified Chebyshev Polynomials of First-Kind
In this section, we will construct the operational matrix of derivatives of Robin-Modified Chebyshev polynomials of first-kind n (x), n = 0, 1, 2, … .To do that, we need to extract the first derivative of n (x) in terms of these polynomials themselves.First, we can see that This leads us to state and prove the main theorem, by which a novel Galerkin operational matrix of derivatives will be introduced.

Theorem 4.1
The first derivative of n (x) for all n ≥ 0 , can be written in the form where the expansion coefficients a 0 (n), a 1 (n) … , a n−1 (n) , are the solution of the system where The elements of G n and B n are defined as follows: And the two coefficients e 0 (n) and e 1 (n) have the form: Proof It is not difficult to show that the two coefficients e 0 (n) and e 1 (n) has the form (4.3).So the expansion (4.1) can be written in the form 3) e 0 (n) = (1)  n (0) − Using the two formulae of Maclaurin series for j (x) and D n (x) , with taking into consideration that they are two polynomials of degree (j + 2) and (n + 1) , respec- tively, Eq.(4.4) can be written as, This gives the following triangle system of n equations in the unknown expansion coefficients a 0 (n), which can be written in the matrix form (4.2) and this completes the proof of Theorem 4.1.◻ Now, we have reached the main desired result in this section, that is the operational matrix of derivatives of which is given in the following corollary as a direct consequence of Theorem 4.1.

Corollary 4.1
The mth derivative of the vector (x) has the form: , where

A Collocation Algorithm for Handling Second-Order Differential Equation Subject to Robin Boundary Conditions
In this section, we utilize the operational matrix derived in Corollary 4.1 to get numerical solutions for the second-order BVP (1.1)-(1.3).

Homogeneous Boundary Conditions
Suppose that the boundary conditions (1.2) and (1.3) are homogeneous, that is, We can consider an approximate solution to y(x) in the form where Corollary 4.1 enables us to approximate the derivatives y (m) (x), m = 1, 2 in matrix form: In this method, using the approximations (5.1) and ( 5.

2) allow one to write the residual of equation (1.1) as
To obtain the numerical solution of the equation (1.1) subject to the two conditions (1.2) and (1.3) (with 1 = 2 = 0 ), a spectral approach is suggested in the cur- rent section: the Robin shifted Chebyshev first-kind collocation operational matrix method RSC1COMM.The collocation points are the (N + 1) zeros of T * N+1 (x), (4.9) (5.1) so we have then the unknown coefficients c i (i = 0, 1, ..., N) can be obtained by solving (N + 1) linear or nonlinear algebraic equations (5.4) using any suitable solver.

Nonhomogeneous Boundary Conditions
An important step in constructing the suggested algorithm is converting the equation (1.1) with respect to non-homogeneous Robin conditions (1.2) and (1.3) into the corresponding homogeneous conditions.To do that, we use the following transformation: Hence it suffices to solve the following modified equation subject to the homogeneous Robin conditions Then

Convergence and Error Estimates For RSC1COMM
In this section, the convergence and error estimates of the suggested approach are examined.For a positive integer N, consider the space S N defined by Additionally, the error between y(x) and its approximation y N (x) is defined by In the paper, the error of the numerical scheme is analyzed by using: (5.9) y N (x) = ȳN (x) + x + .
1 3 Journal of Nonlinear Mathematical Physics (2023) 30:1239-1263 The L 2 norm error estimate, and the L ∞ norm error estimate, The proof of the following theorem is similar to the proofs of theorems presented in the research papers [ where Additionally, we have then we can deduce the following two inequalities: and some algebraic computations, the two inequalities (6.4) and (6.5) lead to the two estimates(6.14)and (6.15), and this completes the corollary's proof.◻ (6.9) (2(N + 1) + 1) .
Problem 7.5 Consider the nonlinear boundary value problem, [9,10,35] subject to the non-homogeneous Robin conditions The exact solution is y(x) = ln(x + 1) and the computed approximate solution y 17 (x) has the form: This solution agrees perfectly with the exact solution of accuracy 10 −16 as shown in Table 7.  Problem 7.6 Consider the nonlinear boundary value problem, [9,10,12] subject to the non-homogeneous Robin conditions The exact solution is y(x) = −2ln cos( 2 x − 4 ) − ln( 2) and the computed approxi- mate solution y 19 (x) has the form: This solution agrees perfectly with the exact solution of accuracy 10 −13 as shown in Table 9.
Problem 7.7 Consider the nonlinear boundary value problem [9,11,35], subject to the non-homogeneous Robin conditions The exact solution is y(x) = 2 2 − x − x − 1 and the computed approximate solution y 19 (x) has the form: This solution agrees perfectly with the exact solution of accuracy 10 −16 as shown in Table 11.

Conclusion
Herein, a system of modified shifted Chebyshev polynomials of the first kind that satisfies homogeneous two boundary Robin conditions has been established.The employment of these polynomials with the collocation spectral method provides an approximation of the given second-order differential equation.The proposed method RSC1COMM was tested using seven examples, which demonstrate the algorithm's high accuracy and efficiency.We believe that the theoretical results in this paper can be utilized to treat other types of ordinary and fractional differential equations.Also, the theoretical convergence and error analysis were discussed, and it was demonstrated that the dependence of error on N when RSC1COMM is employed.The presented numerical problems demonstrated the method's applicability, effectiveness, and accuracy.

n
(t), (,  > −1) .More definitely, we have where (a) n = Γ(a + n)∕Γ(a) is the Pochhammer's Symbol.We defined the so-called shifted Chebyshev polynomials by introducing the change of variable t = 2x−a−b b−a .Let the shifted Chebyshev polynomials T n ( 2x−a−b b−a ) be denoted by T * n (x;a, b) , then In this respect, the orthogonality relation for the modified Chebyshev polynomials is where w

andCorollary 6 . 1
Since the approximate solution y N (x) ∈ S N represents the best possible approxima- tion to y(x), we have, as a result, and Employing in particular f (x) = Y N (x) in the previous two inequalities (6.10) and (6.11) leads to the two estimates: and respectively, and the proof is complete.◻The following corollary shows that the obtained error has a very rapid rate of convergence.For all N ≥ 1 , the following two estimates hold: and Proof Making use of the asymptotic result in[33, p.233],

Fig. 12 Log
Fig. 12 Log Errors for Example 7.7

Table 1
Maximum absolute error of Example 7.2

Table 7
Maximum absolute error of Example 7.5

Table 9
Maximum absolute error of Example 7.6

Table 10
Fig. 1 Approximate solution y 11 (x), and Errors E N (x) using various N for Example 7.2