Convergence and Ulam-Hyers-Rassias Stability Analyses of Numerical Solutions of Bratu Type Equations using Picard Method

1. INTRODUCTION

Mathematical modeling of many physical phenomena is achieved in the form of integral equations, differential equations (DEs), or integro- differential equations [1-3]. Bratu type DEs have many applications in practical life. These include fuel ignition, Chandrasekhar model, nanotechnology, thermal reaction, heat transfer by conduction, radiative heat transfer, and chemical reactor theory [4-7]. The scientists and researchers are very keen in the study of Bratu type equations (BTEs). This is because the solutions of such type of equations are more natural and significant due to their important usage in engineering and scientific experimentations. The following equation is called one dimensional Bratu equation [8, 9]

𝑈^′′(𝑥) + 𝜇  𝑒^𝑈(𝑥) = 0      with 𝑈(0) = 0, 𝑈(1) = 0 ;      ∀     𝑥    ∈

[ 0, 1]                                                                                                    (1)

where 𝜇 > 0 is a physical parameter. Exact solution of above equation (1) is10]

Several analytical, semi-analytical, and numerical schemes have been used to solve BTEs. Decomposition Method (DM) [11], Homotopy Perturbation Method (HPM) [12, 13], Perturbation Iteration Algorithm (PIA) [14], Optimal Homotopy Asymptotic Method (OHAM) [15], Laplace Method (LM) [16], Variational Iteration Method (VIM) [17], Perturbation Method (PM) [18], and B Spline Method (BSM) [19] have been implemented to handle BTEs.

While the above-mentioned approaches have been used successfully, they require intricate calculations, for instance, the formation of Adomian

polynomials or determining auxiliary parameters and functions in homotopy-based approaches. The Picard iterative scheme is an attractive alternative considering its simplicity of concept, ease of implementation, and well-established rapid convergence for many nonlinear problems [20-24]. However, its use on the Bratu equation, especially with an extensive study on the stability of its solutions, is an uncharted territory. Therefore, the current article aimed to fill that gap. The novelty of this research is two-pronged: (i) systematic use and numerical verification of the Picard method for BTEs and (ii) pioneering Ulam-Hyers-Rassias stability study of the approximate solutions developed by the iterative scheme.

The study of Ulam-Hyers type stability of various dynamical systems has received considerable attention. The famous Ulam-Hyers stability instigated in a seminar at the Wisconsin University in 1940, where Ulam

[25] suggested the problem of the stability for functional equations. A significant development occurred in 1941, when Hyers [26] gave a partial solution to Ulam’s problem. After an in-depth analysis of the Ulam-Hyers stability structure, some researchers extended and used the concept of Ulam-Hyers stability, such as generalized Ulam-Hyers stability [27, 28], Ulam-Hyers-Rassias stability [29], and generalized Ulam-Hyers-Rassias stability. Huang et al. [30] analyzed Hyers-Ulam stability, while Qarawani

[31] investigated Hyers-Ulam-Rassias stability of nonlinear DEs. Wang et al. considered Ulam-Hyers stability of fuzzy fractional DEs with delay [32, 33].

The foremost aim of the current article is to compute approximate solutions of BTEs using Picard’s method which are new and not described before in the literature. The article is organized as: Section 2 covers conditions for existence and uniqueness of solutions, while section 3 comprehends the illustration of Picard method. Section 4 presents the solutions of numerical examples. The convergence and stability analyses of the derived solutions have been performed in section 5 and 6, respectively. Estimation of error and comparison of results are incorporated in section 7, while conclusion of the article is drafted in section 8.

2. CONDITIONS FOR EXISTENCE AND UNIQUENESS OF SOLUTIONS OF BRATU EQUATIONS

The existence and uniqueness of solutions for the Bratu problem are well-established  and  depend  critically  on  the  parameter  𝜇 .  The

foundational result is summarized in the following theorem:

Theorem 1. There are three cases for μ given in equation (1)

• If 𝜇 = 𝜇₀, then there exists one solution,
• If 𝜇 < 𝜇₀, then there exist two solutions,
• If 𝜇 > 𝜇₀, then no solution

Here 𝜇₀ is called the critical value [12, 13] given as:

1 =1/4� 2 𝜇₀ sinh (4),

and its numerical value is

𝜇₀ = 3.513830719.

Figure 1. Comparison of 𝜇 and 𝖯.

Figure 1 shows that when 𝜇 < 3.513830719, then 𝖯 has two values for each value of 𝜇. Hence, there are two solutions for each

𝜇 < 3.513830719. For 𝜇 = 3.513830719, only one value of

𝖯 = 4.798645359 exists and there is a unique solution. When

𝜇 > 3.513830719, there is no resultant value of 𝖯, and in this case, no solution of Bratu equation exists [12, 13].

3.  DESCRIPTION OF PICARD METHOD

This section discusses the description of Picard method for second- order ordinary differential equation (ODE) and its implementation to BTE.

that is Picard’s iterative scheme for second-order ODE [21, 22].

From the above equation (6), we have [12, 34]

which is the exact solution of equation (10). Taking 𝑈₀(𝑥) = 0 and applying Picard method, we get

𝑈1(𝑥) = 0.5493527280 𝑥 − 0.5 𝑥² ,

𝑈2(𝑥) = 0.0003720238095 𝑥⁸ − 0.001634978357 𝑥⁷ − 0.001651763169 𝑥⁶

+0.01235224910 𝑥⁵ + 0.02909214918 𝑥⁴ − 0.09155878800 𝑥³ − 0.5 𝑥²

+0.5493527280 𝑥,

and

U₃ (x) = 0.5493527280 x −     0.5 x² −    0.09155878800 x³ + 0.02909214918 x⁴ + 0.0183117576 x⁵ −                                      0.003459802640 x⁶ −

0.001764607014  x⁷ +    0.9322482480   ×    10⁻⁴x⁸    +

1.580853532   ×    10⁻⁴x⁹    +    0.4533942348   ×    10⁻⁵x¹⁰    −

1.393134891 ×    10⁻⁵x¹¹    + 0.06120893625 ×    10⁻⁶x¹²    +

0.06540389326   ×    10⁻⁵x¹³ + +  0.04400289   ×    10⁻⁶x¹⁴    −

0.03474251   ×    10⁻⁶x¹⁵    −    0.03008674   × 10⁻⁷x¹⁶    +

0.02236216459 ×    10⁻⁷  x¹⁷ − 0.02261465929   ×    10⁻⁸  x¹⁸

(13)

that is the approximate solution of equation (10).

which is the exact solution of equation (14). Taking 𝑈₀(𝑥) = 0 and applying Picard method, we get

𝑈₁(𝑥) = 1.248217518 𝑥 − 𝑥²  ,

𝑈₂(𝑥) = 0.005952380952 𝑥⁸ − 0.02971946472 𝑥⁷

+0.01860156574 𝑥⁶              + 0.0924087264 𝑥⁵ + 0.0368294189 𝑥⁴

−0.416072506  𝑥³ − 𝑥² + 1.248217518 𝑥,                               (16)

and

𝑈3(𝑥) = −0.115787055 × 10⁻⁶ 𝑥¹⁸ + 0.130074688 × 10⁻⁵ 𝑥¹⁷ − 0.460289 × 10⁻⁵ 𝑥¹⁶

−0.02644413 × 10⁻⁶ 𝑥¹⁵ + 0.2586929702 × 10⁻⁴ 𝑥¹⁴ + 0.2374648306 × 10⁻⁴ 𝑥¹³

−0.1722400368 × 10⁻³ 𝑥¹² − 0.5966016460 × 10⁻³ 𝑥¹¹ + 0.001944801524 𝑥¹⁰

+0.003173141654𝑥⁹ − 0.00655987510 𝑥⁸ − 0.02640249326 𝑥⁷ − 0.001165361882 𝑥⁶

+0.1664290024 𝑥⁵ + 0.03682941898 𝑥⁴ − 0.4160725060 𝑥³ −  𝑥² + 1.248217518 𝑥 ,

(17)

that is the approximate solution of equation (14).

Thus,

cosh ([𝑥 − 0.5] 4.798645359)

𝑈(𝑥) = −2ln (cosh ([𝑥 − 0.5] 4.798645359/ 2 )) ,

                                cosh (4.798645359/4)

⇒ 𝑈(𝑥) = −2 ln(0.5524420911cosh(2.399322680 𝑥 − 1.199661340)),

which is the exact solution of equation (18). Taking 𝑈₀(𝑥) = 0 and applying Picard method, we get

𝑈1(𝑥) = 3.999916982 𝑥 − 0.5 𝑥² ,

𝑈2(𝑥) = 0.00130722869 𝑥⁸ − 0.04183044990 𝑥⁷ + 0.4538503540 𝑥⁶ − 1.5225505 𝑥⁵

−2.196046963 𝑥⁴ − 2.342505194 𝑥³ − 1.756915360 𝑥² + 3.999916982 𝑥 ,

𝑈3(𝑥) = −2.405744959 × 10⁻³𝑥¹⁸ + 6.295431774 × 10⁻³𝑥¹⁷ + 6.06359428 × 10⁻³𝑥¹⁶

−0.04140070848 𝑥¹⁵ + 0.2133926181 𝑥¹⁴ − 0.3033346185 𝑥¹³ − 0.1635341992 𝑥¹²

−0.2943933311 𝑥¹¹ − 1.238108848 𝑥¹⁰ − 0.09860541020 𝑥⁹ + 0.06549570825 𝑥⁸

+1.569248472 𝑥⁷ + 2.820111160𝑥⁶ − 0.2276930256 𝑥⁵ − 1.827997980 𝑥⁴

−2.342505194 𝑥³ − 1.756915360 𝑥² + 3.999916982 𝑥,                              (20)

and

𝑈4(𝑥) = 3.999916982 𝑥 − 1.756915360 𝑥² − 2.342505194 𝑥³  −

2.608816838 𝑥⁴  + 2.057791678 𝑥⁵ + 1.436367948 𝑥⁶ +

0.3755924260 𝑥⁷ − 0.5989937866 𝑥⁸ − 1.115268983 𝑥⁹ −

0.1861262692  𝑥¹⁰ + 0.4014073533𝑥¹¹ + 0.4895580669𝑥¹²

−0.2133778426𝑥¹³         −  0.1783860287𝑥¹⁴ − 0.2345269362𝑥¹⁵ −

0.110694331 𝑥¹⁶ + 0.0325434749  𝑥¹⁷ + 0.0946829798 𝑥¹⁸ +

0.0545570354 𝑥¹⁹ + 0.023638545𝑥²⁰ − 0.0242938671 𝑥²¹ −

0.021804040  𝑥²² + 0.0072424414  𝑥²³ + 0.0741606236  𝑥²⁴ +

0.002463820  𝑥²⁵ − 0.0059534466  𝑥²⁶ − 0.0012045626  𝑥²⁷ −

0.0067346123  𝑥²⁸ − 0.0011077633  𝑥²⁹ + 0.00771358043𝑥³⁰ +

0.000802098𝑥³¹ + 0.0003443211𝑥³² + 0.61416749 × 10⁻⁵  𝑥³³ −

0.0201121798 × 10⁻⁵  𝑥³⁴ − 0.0922424728 × 10⁻⁵  𝑥³⁵ −

0.16190965120 × 10⁻⁶  𝑥³⁶ + 0.056308070 × 10⁻⁶  𝑥³⁷  + 0.00429760 ×

10⁻⁷  𝑥³⁸ + 0.0748094200 × 10⁻⁷  𝑥³⁹ − 0.0306817000 × 10⁻⁷𝑥⁴⁰  −

0.03965400 × 10⁻⁸  𝑥⁴¹ + 0.05616 × 10⁻⁹  𝑥⁴² − 0.01964754709 ×

10⁻⁹  𝑥⁴³ + 0.041984642179 × 10⁻⁹  𝑥⁴⁴ − 0.064080398 × 10⁻¹⁰  𝑥⁴⁵ +

0.07435209 × 10⁻¹¹  𝑥⁴⁶  − 0.067626903 × 10⁻¹²  𝑥⁴⁷ + 0.0489542041 ×

10⁻¹³  𝑥⁴⁸ − 0.0283546 × 10⁻¹⁴  𝑥⁴⁹ + 0.013108496 × 10⁻¹⁵  𝑥⁵⁰ −

0.0047865113 × 10⁻¹⁶  𝑥⁵¹ + 0.013534696 × 10⁻¹⁷  𝑥⁵² − 0.0286394 ×

10⁻¹⁹  𝑥⁵³ + 0.04273016 × 10⁻²³  𝑥⁵⁴ − 0.40268047 × 10⁻²⁵  𝑥⁵⁵ +

0.17958767 × 10⁻²⁷ 𝑥⁵⁶.                                                             (21)

that is the approximate solution of equation (18).

Consider

𝑈^′′(𝑥) + 5  𝑒^𝑈(𝑥) = 0                  with          𝑈(0) = 0 ,          𝑈(1) =

0 ;        ∀        𝑥    ∈    [ 0 , 1].                                                              (22)

Here, 𝜇 = 5. To find the solutions of above equation (22)

𝑈(𝑥) = −2ln (cosh(|𝑥-0.5|P/2)/cosh(p/4)

and 𝖯 can be derived from

𝖯 = √2 × 5 cosh (P/4).

Since 𝜇 = 5, then

𝖯 = √2 × 5 cosh (P/4)                                                                          (23)

After solving equation (23), we get

𝖯 = (−3.181805553 + 5.282305940     𝑖 )√10.

Since 𝖯 is a complex number, therefore it can not be continued further. It has already been deliberated upon that for 𝜇 > 3.513830719, Bratu equation has no solution.

5.  CONVERGENCE ANALYSIS

This section discusses the convergence of solutions of numerical examples derived by Picard method.

The convergence of the sequence 𝑈𝑛(𝑥) generated by Picard’s iteration

is assessed using the uniform convergence criterion on the interval 𝑥 ∈ [𝟎𝟎, 1]. Recall that a sequence of functions 𝑈𝑛(𝑥) converges uniformly to a function 𝑈(𝑥) if [35]

lim 𝑠𝑢𝑝|𝑈_𝑛(𝑥) − 𝑈(𝑥)| = 0.

𝑛→∞

It will be demonstrated numerically that for given examples, the maximum absolute error |𝑈_𝑛(𝑥) − 𝑈(𝑥)| indeed decreases below a predefined tolerance 𝜀𝜀 after a finite number of iterations 𝑛, providing strong empirical evidence for uniform convergence.

Since it has been discussed above that 𝑈𝑛(𝑥) is uniformly convergent if and only if

sup|  𝑈_𝑛(𝑥) − 𝑈(𝑥)| = ∥ 𝑈_𝑛(𝑥) − 𝑈(𝑥) ∥ < 𝜀𝜀

when 𝑚 < 𝑛 , ∀ 𝑥 ∈ [ 0 , 1] .

Taking 𝜀𝜀 = 0.00009, we have

sup|  𝑈_𝑛(𝑥) − 𝑈(𝑥)| = ∥ 𝑈_𝑛(𝑥) − 𝑈(𝑥) ∥ = Max     Error   < 𝜀𝜀

when          𝑚 <    𝑛 ,          ∀        𝑥    ∈ [ 0 , 1]

which implies that 𝑈𝑛(𝑥) converges to 𝑈𝑛(𝑥) , where 𝑚 = 2 and 𝑈(𝑥)

represents exact solution.

Since it has been discussed above that 𝑈𝑛(𝑥) is uniformly convergent if and only if

sup|  𝑈_𝑛(𝑥) − 𝑈(𝑥)| = ∥ 𝑈_𝑛(𝑥) − 𝑈(𝑥) ∥ < 𝜀𝜀

when     𝑚 <    𝑛 ,          ∀ 𝑥 ∈ [ 0 , 1] .

Taking  𝜀𝜀  =      0.0009, we have

sup|  𝑈_𝑛(𝑥) − 𝑈(𝑥)| = ∥ 𝑈_𝑛(𝑥) − 𝑈(𝑥) ∥ = Max     Error   < 𝜀𝜀

when       𝑚 <    𝑛 ,          ∀        𝑥    ∈ [ 0 , 1]

which implies that 𝑈𝑛(𝑥) converges to 𝑈(𝑥) , where 𝑚 = 2 and 𝑈(𝑥)

represents exact solution.

Since it has been discussed above that 𝑈𝑛(𝑥) is uniformly convergent if

and only if

sup|  𝑈_𝑛(𝑥) − 𝑈(𝑥)| =  ∥ 𝑈_𝑛(𝑥) − 𝑈(𝑥) ∥ <    𝜀𝜀

when                 𝑚 <    𝑛 ,          ∀     𝑥    ∈ [ 0 , 1] .

Taking          𝜀𝜀     =      0.022414, we have

sup|  𝑈_𝑛(𝑥) − 𝑈(𝑥)| = ∥ 𝑈_𝑛(𝑥) − 𝑈(𝑥) ∥ = Max     Error   < 𝜀𝜀

when        𝑚 <    𝑛 ,          ∀        𝑥    ∈ [ 0 , 1]

which implies that 𝑈𝑛(𝑥) converges to 𝑈(𝑥), where 𝑚 = 3 and 𝑈(𝑥) is the exact solution.

6.  STABILITY ANALYSIS

This section discusses the stability analysis of numerical scheme for BTEs using the following stability criteria.

The general second-order differential equation is considered as

𝜉^′′(𝑥) −    𝐺(𝑥  , 𝜉(𝑥)  , 𝜉 ^′(𝑥)) = 0 with 𝜉(0) =

0     and 𝜉 ^′(0) = 0.  (24)

Following [29-31], it can be said that equation (24) is Ulam-Hyers-Rassias (UHR) stable with respect to a continuous functions 𝜉(𝑥) if

1. | 𝜉^′′(𝑥) −    𝐺(𝑥  , 𝜉(𝑥)  ,  𝜉 ^′(𝑥))| ≤ 𝐻(𝑥)  ,
2. | 𝜉(𝑥) −    𝜁(𝑥)| ≤ 𝑀(𝑥)  ,

where    𝜉(𝑥) , 𝜁(𝑥) ∈ 𝐶 ²[𝑎, 𝑏]      and    𝐻(𝑥), 𝑀(𝑥)     are    positive continuous functions. Furthermore,  𝐻(𝑥) does not depend on 𝜉(𝑥) and

𝐺(𝑥  , 𝜉(𝑥) , 𝜉 ^′(𝑥)).

Theorem 2. The Bratu equation is Ulam-Hyers-Rassias stable, if it fulfills the conditions given below

1. � 𝑈_𝑛^′′(𝑥) +    𝜇𝑒^𝑈𝑛(𝑥)� ≤ 𝐻(𝑥) ,
2. | 𝑈_𝑛(𝑥) −    𝑈(𝑥)| ≤ 𝑀(𝑥).

While a full theoretical proof is beyond the scope of this numerical study, strong numerical evidence will be provided for the stability of derived approximate solutions. For Picard iterations 𝑈𝑛(𝑥) , it will be numerically verified that these inequalities hold for given examples.

Consider

𝑈^′′(𝑥) + 𝑒^𝑈(𝑥) = 0      with               𝑈(0) = 0 ,             𝑈(1)

= 0 ;                     ∀        𝑥    ∈    [ 0 , 1].

Here,  𝜉(𝑥) = 𝑈₃(𝑥)  and  𝜁(𝑥) = 𝑈(𝑥), then

� 𝑈₃^′′(𝑥) + 𝑒^𝑈3(𝑥)� ≤ max� 𝑈₃^′′(𝑥) + 𝑒^𝑈3(𝑥)� ≤ 𝐻(𝑥)

and

= 𝑥² + 0.0038857183 ,

| 𝑈3(𝑥) − 𝑈(𝑥) |        ≤  max| 𝑈3(𝑥) − 𝑈(𝑥) | ≤  𝑀(𝑥)

= 𝑥² + 0.000089546 .

Hence, Example-1 fulfills the hypothesis of Theorem 2 and is Ulam-Hyers-Rassias stable.

Consider

𝑈^′′(𝑥) + 2  𝑒^𝑈(𝑥) = 0        with        𝑈(0) = 0 ,             𝑈(1)

= 0 ;             ∀        𝑥    ∈    [ 0 , 1].

Here,  𝜉(𝑥) = 𝑈₃(𝑥)  and  𝜁(𝑥) = 𝑈(𝑥), then

� 𝑈₃^′′(𝑥) + 2 𝑒^𝑈3(𝑥)� ≤ max� 𝑈₃^′′(𝑥) + 2 𝑒^𝑈3(𝑥)� ≤ 𝐻(𝑥)

and

= 𝑥⁴ + 0.09701083186 ,

| 𝑈3(𝑥) − 𝑈(𝑥) |        ≤  max| 𝑈3(𝑥) − 𝑈(𝑥) |  ≤  𝑀(𝑥)

= 𝑥⁴ + 0.00082169 .

Thus, Example-2 follows the hypothesis of Theorem 2 and is Ulam-Hyers-Rassias stable.

Consider

𝑈^′′(𝑥) + 3.513830719 𝑒^𝑈(𝑥) = 0       with      𝑈(0) = 0 ,          𝑈(1)

= 0 ;  ∀ 𝑥     ∈ [ 0 , 1].

Here,  𝜉(𝑥) = 𝑈₄(𝑥)  and  𝜁(𝑥) = 𝑈(𝑥), then

� 𝑈₄^′′(𝑥) + 3.513830719 𝑒^𝑈4(𝑥)� ≤ max� 𝑈₄^′′(𝑥) + 3.513830719 𝑒^𝑈4(𝑥)�

≤  𝐻(𝑥) = 𝑥⁶ + 65.60872438 ,

and

|𝑈₄(𝑥) − 𝑈(𝑥) |  ≤  max| 𝑈₄(𝑥) − 𝑈(𝑥) |  ≤  𝑀(𝑥)

= 𝑥⁶ + 0.022414  .

Thus, Example-3 fulfils the hypothesis of Theorem 2 and is Ulam- Hyers-Rassias stable.

7.  RESULTS AND DISCUSSION

Here, the efficacy and veracity of Picard’s iterative scheme has been demonstrated through error’s estimation, comparison of maximum errors, and graphical illustrations of the obtained approximate solutions of BTEs.

Table 1 contains |E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄| estimated using Picard method, which shows that 𝐦𝐦𝐦𝐦𝐦𝐦 |E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄| = ∥ E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄 ∥ = 𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎. The ∥ E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄 ∥ estimated by Picard method, PIA [21], DM [18], HPM [13], and OHAM >[15] are listed in Table 2. It explained that Picard’s method has smallest ∥ E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄 ∥.

Figure 2(a , b) demonstrates the comparison of exact solution with approximate solutions obtained by Picard’s iterative scheme. Figure 2(a) shows that both approximate and exact solutions coincide with each other. Figure 2(b) indicates that approximate solutions of zero to third orders are converging to exact solution. Figure 2(c) illustrates the comparison of solutions obtained by Picard’s iterative scheme, HPM, OHAM, DM, and PIA with the exact solution. This demonstrates that both exact and Picard’s solutions make an excellent agreement, while other solutions diverge from exact solution. Figure 2(d) indicates that Picard’s iterative scheme has least ∥ Error ∥ as compared to other stated techniques.

Table 1. Error Estimation of Example-1

Table 2. Comparison of  Error    of Example-1

Figure 2. Graphical Illustration of Example-1

Table 3 lists | E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄 | estimated by Picard’s iterative scheme. It can be observed that ∥ E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄 ∥  = 𝐦𝐦𝐦𝐦𝐦𝐦 | E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄 | = 𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎1𝟎𝟎𝟎𝟎. The ∥ E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄 ∥ estimated by Picard method, Laplace method (LM) [16], DM [18], PIA [14], and HPM [13] drafted in Table 4 indicates that Picard’s method has smallest ∥ E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄 ∥ amongst these methods.

Figure 3(a , b) illustrated the comparison of exact solution with approximate solutions obtained by Picard’s iterative scheme. Figure 3(a) shows that both approximate and exact solutions match with each other. Figure 3(b) shows that approximate solutions of zero to third orders are converging to exact solution. Figure 3(c) demonstrates the comparison of solutions obtained by Picard’s iterative scheme, DM, LM, PIA, and HPM with the exact solution. This illustrates that both exact and Picard’s solutions make an excellent agreement, while solutions derived by other mentioned techniques diverge from exact solution. Figure 3(d) represents that Picard’s iterative scheme has least ∥ Error ∥ as compared to other stated methods.

Table 3. Error Estimation of Example-2

Table 4. Comparison of   Error     of Example-2

Figure 3. Graphical Illustration of Example-2

The | E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄 | estimated by Picard’s iterative scheme is drafted in Table  5  which  shows  that  ∥ E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄 ∥  = 𝐦𝐦𝐦𝐦𝐦𝐦 | E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄 | =𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎1𝟎𝟎. The ∥ E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄 ∥estimated by Picard method, B Spline method [19], HPM [13], PIA [14], and VIM [17] are drafted in Table 6 which shows that Picard method has smallest ∥ E𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄 ∥ amongst these methods.

Figure 4 (a, b) represents the comparison of exact solution with approximate solutions obtained by Picard’s iterative scheme. Figure 4(a) shows that both approximate and exact solutions match with each other. Figure 4(b) shows that approximate solutions of zero to fourth orders are converging to exact solution. Figure 4(c) illustrates a comparison of approximate solutions computed by Picard’s iterative scheme, BSM, HPM, PIA, and VIM with exact solution. This demonstrates that both exact and Picard’s solutions make an excellent agreement, while solutions computed by other mentioned methods move away from exact solution. Figure 4(d) displays that Picard’s iterative scheme has smallest ∥ Error ∥ as compared to other stated schemes.

Table 5. Error Estimation of Example-3

Figure 4. Graphical Illustration of Example-3

Table 6. Comparison of   Error     of Example-3

8.  Conclusion

The current article is devoted to the derivation of approximate solutions of BTEs using Picard’s iterative technique which is a reliable, effective, and efficient method to solve such kind of equations. Existence and uniqueness of solutions have been confirmed. Convergence and stability of solutions of numerical examples have also been examined. Furthermore, the absolute errors for all examples have also been estimated. The Picard’s results are compared with the results attained by DM, PIA, HPM, OHAM, LM, VIM, PM, and BSM through tables and graphical illustrations, and then discussed thoroughly. The derived results definitely designate the following outcomes: a) The solutions of BTEs are convergent, b) The solutions of BTEs are Ulam-Hyers-Rassias stable, c) Picard’s solutions have least ∥Error∥ amongst other mentioned techniques, and d) Picard’s solutions are in good agreement with exact solutions.

The results of the current research open up a number of promising directions for extension. The Picard’s iterative method can be extended to solve higher-dimensional Bratu problems, systems of Bratu equations, and problems with nonlinear boundary conditions. In addition, investigating its application to fractional-order Bratu type equations involving Caputo, Riemann-Liouville, or other fractional derivatives is an important and difficult direction for future study. The stability framework developed here can also be used for other classes of nonlinear differential equations that are solved using iterative methods.

Author Contribution

Saif Ullah: conceptualization, formal analysis, investigation, methodology, project administration, resources, supervision, validation, writing – original draft, writing – reviewing & editing.

Muzaher Ali: data curation, formal analysis, investigation, methodology, software, visualization, writing – original draft, writing – reviewing & editing.

Sana Bajwa: data curation, formal analysis, investigation, methodology, software, visualization, writing – original draft, writing – reviewing & editing.

Ahsan Bilal: data curation, formal analysis, investigation, methodology, writing – original draft, writing – reviewing & editing.

Conflict of Interest

The authors of the manuscript have no financial or non-financial conflict of interest in the subject matter or materials discussed in this manuscript.

Data Availability Statement

The data associated with the study may be provided by the corresponding author if requested.

Funding Details

No funding was received for this research.

Generative AI Disclosure Statement

The authors did not use any type of generative artificial intelligence software for this research