A fixed point iterative scheme based on Green's function for numerical solutions of singular BVPs

1.   Introduction

In some sense, all the real world phenomena can be essentially modeled in the form of differential equations having certain boundary conditions ^[1]. This is one of the reasons that the study of differential equations is too important. One of the difficulties that arise naturally in studying differential equations is that their sought solutions are explicitly unknown (or it is very hard to solve them using available analytical approaches). In this case, the sought solution is thus reasonably possible to set the form of a fixed point problem of an operator (whose domain is possibly some distance space). However, when a fixed point of this operator exists, then one naturally thinks how its approximate value can be computed using an appropriate numerical scheme. In 1922, Banach ^[2] proved that if the domain of such an operator is complete normed space and the operator is contraction, then such operators admit a unique fixed point (which is the unique solution of the underlying problem) and the sequence of Picard ^[3] iterates essentially converge to this unique fixed point. This result has many useful applications in differential and integral equations because it gives the existence and approximation of a solution for these problems. Notice that if $B$ denotes a Banach space with the norm $||.||$, then the operator $F:B\rightarrow B$ is called a contraction (sometimes called a Banach-contraction) if for all $v, w\in B$ it is possible to find a real constant $\mu\in[0, 1)$, such that

We say that a point $s^{\ast}\in B$ is known as a fixed point of $F$ when the equation $Fs^{\ast} = s^{\ast}$ holds and we write $fix(F)$ to denote a set of all fixed points. In this case, the Picard iteration ^[3] of $F$ is defined as:

We know that the selfmap $F$ is known as a nonexapnsive mapping on $B$ if the relation (1.1) holds for $\mu = 1$. Fixed point approximation under different iterative schemes is an active and important area of research on its own ^[4,5,6,7]. Browder's ^[8] (cf. also Gohde ^[9] and Kirk ^[10]) fixed point theorem suggests a fixed point (may not unique) for a certain nonexpansive operator in a Banach space setting. Moreover, there are some well-known numerical examples of nonexpansive operators, for which the Picard iteration is not convergent to its fixed point (see, e.g., ^[11] and others). To overcome the case of nonexpansive mappings, Mann ^[12] suggested a new iteration scheme which needs an initial value as well as a sequence of real numbers whose values are between $0$ and $1$.

The Mann iteration ^[12] recursively generates a sequence as:

where $\alpha_{m}\in[0, 1]$.

In the literature of iterative schemes, it is known that the speed of the both Picard ^[3] and Mann ^[12] iteration is slow. To achieve a better rate of convergence, Khan ^[13] combined the iterative schemes due to Picard and Mann and named the resultant iterative scheme as a Picard-Mann hybrid iterative scheme (PMH-iterative scheme).

Precisely, PMH-iterative scheme ^[13] recursively generates a sequence as:

where $\alpha_{m}\in(0, 1)$.

Khan proved that the PMH-iteration scheme (1.4) essentially converges to a fixed point of a certain operator. Moreover, he proved analytically and numerically that this scheme suggests high accurate results corresponding to Mann and Picard iterations. Existence and approximation of solutions for BVPs is an important area on its own. Different techniques have been studied by authors for existence and approximation of solutions for various classes of BVPs ^[14]. On the other hand, Khuri and Sayfy ^[15,16,17] embedded Green's function into some well-known iterative schemes and proved that these new type of schemes suggest high accurate results corresponding to the other available methods of the literature. Motivated by Khuri and Sayfy, Assadi et al. ^[18] introduced Picard-Green's and Mann-Green's iterative schemes for a class of SBVPs and proved that both of these schemes are better than the many other previous iterative schemes studied for SBVPs. Thus, the challenging question is when is it possible to obtain a modified version of the scheme (1.4) based on Green's function for finding solution of SBVP? In this paper, we first obtain the requested version of this scheme for SBVPs and name it PMH-Green's iterative scheme and show that the PMH-Green produces very high accurate results compared to Mann-Green's and Picard-Green's iteration schemes. We also show that the PMH-Green's iterative scheme is weak $w^{2}$-stable in this case. The numerical computations given at the end of the paper supports the main outcome of the paper and suggests the numerical effectiveness of the proposed scheme.

2.   Overview of the iterative scheme

To propose the desired PMH-Green's iterative scheme, first we define some elementary concepts and results that are necessary for the main work.

2.1.   Construction of the Green's function

This subsection will establish the Green's function for a broad class of SBVPs. To succeed in this aim, suppose $t\in(a, b)$, we consider a linear second order equation which is mathematically written as following:

and the associated boundary conditions (BCs) are the following:

It should be noted that the possible general solution is given as $w(t) = w_{h}(t)+w_{p}(t)$. Here, the function $w_{h}(t)$ is essentially the solution for the equation $L[w] = 0$ subjected to the BCs suggested in (2.2), and $w_{p}(t)$ is solution for the equation $L[w] = f (t)$ endowed with the homogeneous BCs given below:

Now, for finding $w_{p}(t)$, one needs a solution for

which is essentially subject to the BCs as given in (2.3) and in this case, such a solution is known as a Green's function, denoted normally by $G(t, s)$. Then

Let $w_{1}$, $w_{2}$ be two linearly independent solutions of $L[w] = 0$. Notice that the Green's function essentially obeys the homogeneous equation for each choice of $t\neq s$ and thus it will be a linear combination of $w_{1}$ and $w_{2}$:

the constants $c_i$ and $d_{i}$, $(i = 1, 2)$ are determined using the following axioms:

($A_{1}$) $G$ satisfies the given homogeneous BCs, i.e.,

($A_{2}$) Continuity of $G$ at $t = s$:

($A_{3}$) Jump discontinuity of $G^{\prime}$ at $t = s$, i.e.,

For nonlinear SBVPs

the particular solution satisfies

and here $G$ is the Green's function connected to (2.9).

2.2.   PMH-Green's iterative scheme

Now we propose our desired PMH-Green's iterative scheme for approximate solutions of the following SBVPs of the form as follows:

and the associated BCs are as given in (2.2). Now consider the Green's function $G$ corresponding to the linear term and we consider the following operator:

Now from (2.10) and (2.12), we obtain

Put $w_{p} = w$ in (2.13), one has

Hence from (2.14), we obtain the modified form of PMH-iterative scheme given in (1.4) as follows:

which yields the following iterative procedure:

It follows that

and here $L$ denotes the linear term and the initial value to start the scheme, that is, $w_{0}$ must be chosen in a way that satisfies the Eq (2.11), $L[w] = 0$, and the given specified BCs.

3.   Convergence result

We are now interested in establishing the main convergence result. For this, let $q\geq2$ and consider a SBVP as provided below:

and the associated BCs are the following:

To construct the required Green's function associated with (3.1), we apply the axioms of Green's function which we gave in the last section. Hence, after solving the Eq (3.1), one has

Accordingly, using homogenous BCs as given in (3.2), that is, $w^{\prime}(0) = w(1) = 0$, one has the following

Applying the continuity axioms of Green's function, we obtain

Using unit jump discontinuity connected to the first derivative of the Green's function, we have

After solving (3.3)–(3.5), we get the desired Green's function as follows:

Now, embedding the above Green's function in the PMH-iterative scheme given in (2.17), we get the following PMH-iterative scheme:

It follows that

Set $F_{G}:C[0, 1]\rightarrow C[0, 1]$ by

Then (3.7) becomes

The main result of the paper is now ready to establish.

Theorem 3.1. Consider a Banach space $B = C[0, 1]$ with the supremum norm. Let $F_{G}:B\rightarrow B$ be the operator defined in (3.8) and $\{w_{m}\}$ be the sequence of PMH-Green's iterative scheme (3.9). Assume that the following conditions hold:

(a) $\frac{1}{2(q-1)}M_{c} < 1$, where $M_{c} = \max_{[0, 1]\times\mathbb{R}^{2}}|\frac{\partial f}{\partial w}|$.

(b) $\sum\alpha_{m} = \infty$ or for some $\alpha$, $0 < \alpha\leq\alpha_{m}$.

Subsequently, $\{w_{m}\}$ converges strongly to the unique solution of the problems (3.1) and (3.2).

Proof. By using assumption (a), we show that $F_{G}$ is a Banach-contraction, that is, $||F_{G}v-F_{G}w||\leq\mu||v-w||$ for all $v, w\in B$ and some fixed $\mu\in[0, 1)$. To do this, direct integration gives us,

Also, integration by parts twice, we get

Now integrating once, we have

From (3.10)–(3.12), we have

From (3.2), $w(1) = \beta$, therefore (3.13) takes the following form

Hence

Now a simple integration suggests

It is easy to see that the $g(t)$ attains the maximum value in $[0, 1]$ either at endpoints or on the critical points. Hence

By (3.15)–(3.17), one has

But $\mu = \frac{1}{2(q+1)}M_{c} < 1$, it follows from (3.18), that $F_{G}$ is a Banch contraction. Since $B$ is complete and $F_{G}$ is a Banach-contraction, thanks to the BCP ^[2], $F_{G}$ admits essentially a unique fixed point in $B = C[0, 1]$ and this point we shall denote by $s^{\ast}$ and hence it follows that this $s^{\ast}$ is a unique solution of the problems (3.1) and (3.2).

Moreover, from the assumption (b), we will prove that PMH-Green's iterative converges strongly to $s^{\ast}$. First, we assume the case when $\sum\alpha_{m} = \infty$. Now

Hence, we obtain

Finally, using (3.19), we compute $||w_{m+1}-s^{\ast}||$ as follows.

Hence, we obtain

Now, from (3.20), we step by step obtain the following

It follows that

Noting that $\lim_{m\rightarrow \infty}(\mu)^{m} = 0$ because $\mu\in[0, 1)$. Also, it is well-known from the classical analysis that $1-w\leq e^{-w}$ for all $w\in[0, 1]$. Taking these facts into account with (3.21), we get

As supposed $\sum\alpha_{m} = \infty$ and $\mu$ lies in $[0, 1)$, we have from (3.22) that

Accordingly, $\{w_{m}\}$ converges to a fixed point $s^{\ast}$ of $F_{G}$ which is the unique solution of the problems (3.1) and (3.2). The case when $0 < \alpha\leq\alpha_{m}$ is included already in the case (a) and hence omitted. □

4.   Stability

In all branches of mathematics where iterative methods are used for finding approximate value of the sought solution, stability analysis is one of the desirable properties for such schemes (see ^[19,20,21] and others). Fixed point procedure may or may not stable when we implement them on a certain operator equation ^[22] (cf. also ^[23,24] and others). Suppose a given iterative scheme of a certain operator is convergent to some of its fixed point. In this case, the iterative scheme is said to be stable if and only if the estimated error between two successive iterative strategy does not affect its so-called convergence. As many know, stability for fixed points iterations finds its initial roots in the paper due to Urabe ^[25]. Motivated by Urabe ^[25], Harder and Hicks ^[26] constructed mathematical definition for stability. Some basic concepts that we need in the work are recalled below.

Definition 4.1. ^[26] Consider a mapping $F$ of a Banach space $B$ and suppose $\{w_{m}\}\subseteq B$ is a sequence generated from certain iterative scheme using the mapping $F$ as follows:

here, the element $w_{0}$ denotes starting point and $\gamma$ is a function of $F$ and $w_{m}$. Assume that the sequence of iterates $\{w_{m}\}$ converges to $s^{\ast}\in fix(F)$. In this case, $\{w_{m}\}$ is said to be stable if and only if

where $\{s_{m}\}$ is any chosen sequence in the space $B$.

Definition 4.2. ^[27] Suppose $\{s_{m}\}$ and $\{w_{m}\}$ are any two sequences in a Banach space. We say that these two sequences equivalent if and only if $\lim_{m\rightarrow +\infty}||s_{m}-w_{m}|| = 0.$

Opposed to the concept of arbitrary sequences, Timis ^[28], used the concept of equivalent sequences and obtained a new mathematical definition of the weak stability. This new type of stability is called the weak $w^{2}$-stability. The formal definition is given below.

Definition 4.3. ^[28] Consider a Banach space $B$ and $F$ a selfmap on $B$. If $\{w_{m}\}$ is a sequence of iterates of $F$ produced by the formula (4.1). Assume that $\{w_{m}\}$ is convergent to a point $s^{\ast}\in fix(F)$. Then $\{w_{m}\}$ is said to be weak $w^{2}$-stable if for every equivalent sequence $\{s_{m}\}\subseteq B$ of $\{w_{m}\}$, one has the following

Using the above concepts, we now show that our PMH-Green's iterative scheme (3.9) is weak $w^{2}$-stable.

Theorem 4.4. Let $B$, $F_{G}$ and $\{w_{m}\}$ be as given in the Theorem 3.1. The $\{w_{m}\}$ is essentially weak $w^{2}$-stable with respect to $F_{G}$.

Proof. To complete the proof, we consider any equivalent sequence $\{s_{m}\}$ of $\{w_{m}\}$, that is $\lim_{m\rightarrow \infty}||s_{m}-w_{m}|| = 0$. Put

where $r_{m} = (1-\alpha_{m})s_{m}+\alpha_{m}F_{G}s_{m}$.

Assumed that $\lim_{m\rightarrow +\infty}\epsilon_{m} = 0$. First we compute the estimate $||r_{m}-v_{m}||$. For this,

Consequently, we find

Keeping (4.2) in mind, we can proceed as follows:

Subsequently, we obtain

By assumptions, $\lim_{m\rightarrow +\infty}\epsilon_{m} = 0$ and $\lim_{m\rightarrow \infty}||s_{m}-w_{m}|| = 0$ because $\{s_{m}\}$ is an equivalent sequence for $\{w_{m}\}$. Also $\lim_{m\rightarrow +\infty}||w_{m}-s^{\ast}|| = 0$ due to the convergence of $\{w_{m}\}$ towards $s^{\ast}$. Accordingly, from (4.3), $\lim_{m\rightarrow +\infty}||s_{m}-s^{\ast}|| = 0$. This means that $\{w_{m}\}$ generated by PMH-Green's iterative scheme (3.9) is weak $w^{2}$-stable with respect to the mapping $F_{G}$. □

5.   Numerical example and computations

In this section, we consider several numerical examples to show the high accurate numerical results produced by our proposed method.

Example 5.1. First, we consider the following SBVP which represents the equilibrium of isothermal gas sphere ^[29]:

subjected with the BCs:

where $0 < t < 1$. The exact solution of (5.1) and (5.2) is $w(t) = \sqrt{\frac{3}{3+t^{2}}}$. Take $w_{0}(t) = \sqrt{\frac{3}{4}} = 0.866025$ which satisfies the equation $w^{\prime\prime} = 0$ and given BCs.

Now using Example 5.1, the proposed scheme takes the following form:

Now, for $\alpha_{m} = 0.99$, the values generated by Picard-Green's, Mann-Green's and PMH-Green's iterative schemes in Tables 1–3. Clrearly the PMH-Green's iterative scheme moving faster to the solution. While Green's function involved in scheme (5.3) is provided in Figure 1, the graphical comparison of the absolute errors in this case is given in Figure 2.

We finish the section with following example.

Example 5.2. Now we consider a SBVP whose exact solution is not known explicitly as follows:

subjected to the BCs:

where $0 < t < 1$.

The initial iterate $w_{0}(t) = 0$ corresponding to $w^{\prime\prime} = 0$ and given BCs.

Now using Example 5.2, the proposed scheme takes the following form:

Green's function involved in scheme (5.6) is provided in Figure 3. The absolute error in this case is given in Table 4. The graphical comparison of the absolute errors in this case is given in Figure 4. Again, we see that the PMH-Green's iterative approach is more accurate than the Picard-Green's and Mann-Green's iterative approaches for problems (5.4) and (5.5).

6.   Conclusions

We modified the PHM-iterative scheme by embedding a Green's function of a certain SBVP. The convergence to a sought solution to a given SBVP is proved under some possible mild conditions. We proved the proposed iterative scheme is weak $w^{2}$-stable. Some numerical experiments are performed and it has been shown the numerical accuracy of the PMH-Green's is more accurate and corresponds to the Picard-Green's and Mann-Green's iterative schemes studied by Assadi et al. ^[18]. Since the proposed scheme is stable and gives high accurate numerical solutions in the setting of SBVPs, we conclude that our results improve and extend many other results of the literature due to various authors. Our next aim is to use the proposed scheme for other BVPs that arise in nano-fluid and mathematical physics. Eventually, we point out the following:

$({\rm{i}})$ If we prove the mapping $F_{G}$ a Kannan contraction, that is, for all $v, w$, there exists a constant $\mu\in[0, \frac{1}{2})$ such that $||F_{G}v-F_{G}w||\leq\mu[||v-F_{G}v||+||w-F_{G}w||]$, then $F_{G}$ also admits a unique fixed point. Thus, our iterative scheme can be used to approximate the solutions. However, the Kannan mappings are sometimes not continuous ^[30]; therefore, this is not a good option in our case. Furthermore, the space $C[0, 1]$ contains only continuous functions.

$({\rm{ii}})$ Our iterative scheme uses only one sequence of scalars $\{\alpha_{m}\}$ and the condition we imposed on it is very simple in our convergence theorem. Thus, our iterative scheme needs few parameters to start. Moreover, it gives high accurate results for small values of this sequence.

$({\rm{iii}})$ The iterative scheme is proved weak $w^{2}$-stable. We know that every stable iterative scheme is weak $w^{2}$-stable but the converse is not generally true. Thus, our result of weak $w^{2}$-stability contains the case when one proves a stability result for our iterative scheme in the classical sense.

$({\rm{iv}})$ Since the Banach space used in this paper contains only continuous functions, it is a challenge for us to replace this Banach space by a Banach space that also contains some discontinuous functions.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Authors contributions

Muhammad Arshad gave the idea as a supervisor. Junaid Ahmad wrote the initial draft. Reny George edited the final version and approved for submission.

Acknowledgments

The authors extend their appreciation to Prince Sattam bin Abdulaziz University, Saudi Arabia for funding this research work through the project number (PSAU/2023/01/9010).

Conflict of interest

The authors declare that they have no conflicts of interest.