# Generalized Bessel Quasilinearization Technique Applied to Bratu and Lane–Emden-Type Equations of Arbitrary Order

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Concepts

#### 2.1. Fractional Notations

**Definition**

**1.**

#### 2.2. Bessel Functions

#### 2.3. Quasilinearization Approach

## 3. Direct Bessel Method

**Theorem**

**1.**

#### 3.1. Model Problem (a)

#### 3.2. Model Problem (b)

#### 3.3. The Initial and Boundary Conditions in the Matrix Representation Forms

Algorithm 1: The calculation of the $\gamma $-derivative of ${\mathit{\chi}}_{\alpha}\left(t\right)$. |

procedure$\left[{\mathit{\chi}}_{\alpha}^{\left(\gamma \right)}\right]$ = compute_DX$(M,\gamma ,\alpha )$ |

${\mathit{\chi}}_{\alpha}^{\left(\gamma \right)}\left[1\right]:=0$; |

for$\ell :=1,\dots ,M$do; |

if ($\ell \alpha -\gamma \phantom{\rule{3.33333pt}{0ex}}<\phantom{\rule{3.33333pt}{0ex}}0$) then |

${\mathit{\chi}}_{\alpha}^{\left(\gamma \right)}[\ell +1]:=0$; |

else |

if (($\ell \alpha <\lceil \gamma \rceil )$ && ($\ell \alpha -\lfloor \ell \alpha \rfloor ==0$)) then |

${\mathit{\chi}}_{\alpha}^{\left(\gamma \right)}[\ell +1]:=0$; |

else |

$\mathit{\chi}}_{\alpha}^{\left(\gamma \right)}[\ell +1]:=\frac{\Gamma \left(\ell \alpha +1\right)}{\Gamma \left(\ell \alpha +1-\gamma \right)}{t}^{\ell \alpha -\gamma$; |

end if |

end if |

end for |

end; |

## 4. Bessel-QLM

#### 4.1. Quasilinear Model Problem (a)

#### 4.2. Quasilinear Model Problem (b)

#### 4.3. Residual Error Functions

## 5. Error Analysis of Bessel-QLM

**Theorem**

**2.**

## 6. Numerical Simulations

#### 6.1. Model Problem (a)

**Example**

**1.**

**Example**

**2.**

#### 6.2. Model Problem (b)

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The graphs of approximated and exact solutions (

**left**) and the related absolute errors (

**right**) for $\beta =2,\alpha =1$, $M=10$, and $s=5$ in Example 1.

**Figure 2.**The approximated Bessel-QLM series solutions for Example 1 using various $\beta ,$$\alpha =1.5$, 1.6, $\dots ,2$ for $M=10$, $s=5$.

**Figure 3.**The graphs of numerical and exact solutions for $M=8$ (

**left**) and the resulting absolute errors for $M=8,12,16,20$ (

**right**) for $\beta =2,\alpha =1$, $\lambda =-{\pi}^{2}$, and $s=5$ in Example 2.

**Figure 4.**Numerical approximations in Bessel-QLM for various $\beta =2,1.9,\dots ,1.5$, $\alpha =1$, $\lambda =-{\pi}^{2}$, and $M=15$ in Example 2.

**Figure 5.**The graphs of of errors (

**left**) and the resulting residual errors (

**right**) for $\beta =2,\alpha =1$, $M=8$, and $s=5$ in Example 3.

**Figure 6.**Approximated solutions with $\eta =+1$ (

**left**) and $\eta =-1$ (

**right**) in Bessel-QLM for $M=10,\alpha =1$, $s=5$, and various $\beta =1.9,1.8,\dots ,1.5$ in Example 3.

**Figure 7.**The graphs of numerical and exact solutions for $M=10$ (

**left**) and the resulting absolute errors for $M=5,10,15,20$ (

**right**) for $\beta =2,\alpha =1$, and $s=5$ in Example 4.

**Figure 8.**Numerical approximations in Bessel-QLM for various $\beta =2,1.9,\dots ,1.5$, $\alpha =1$, and $M=10$ in Example 4.

**Figure 9.**Absolute errors in Bessel-QLM for $\beta =2$, $\alpha =1,2$ and various $M=5,10,20$ in Example 5.

**Figure 10.**A comparison of numerical solutions in Bessel-QLM for $M=10$ and various $1<\beta =\alpha <2$ in Example 5.

**Table 1.**The comparison of absolute errors using Bessel-QLM in Example 1 for $\beta =2,\alpha =1$, $M=15,20,25,30$, and various $t\in [0,1]$.

t | ${\mathcal{E}}_{15,1}^{\left(6\right)}\left(\mathit{t}\right)$ | ${\mathcal{E}}_{20,1}^{\left(6\right)}\left(\mathit{t}\right)$ | ${\mathcal{E}}_{25,1}^{\left(6\right)}\left(\mathit{t}\right)$ | ${\mathit{u}}_{30,1}^{\left(6\right)}\left(\mathit{t}\right)$ | ${\mathcal{E}}_{30,1}^{\left(6\right)}\left(\mathit{t}\right)$ |
---|---|---|---|---|---|

$0.1$ | $2.9055\times {10}^{-6}$ | $2.5768\times {10}^{-8}$ | $1.1549\times {10}^{-10}$ | $0.010016711246980$ | $5.0955\times {10}^{-13}$ |

$0.2$ | $6.7455\times {10}^{-6}$ | $5.6983\times {10}^{-8}$ | $2.4911\times {10}^{-10}$ | $0.040269546105905$ | $1.0880\times {10}^{-12}$ |

$0.3$ | $1.0727\times {10}^{-5}$ | $8.9392\times {10}^{-8}$ | $3.8795\times {10}^{-10}$ | $0.091383311853805$ | $1.6892\times {10}^{-12}$ |

$0.4$ | $1.4945\times {10}^{-5}$ | $1.2378\times {10}^{-7}$ | $5.3536\times {10}^{-10}$ | $0.164458038152439$ | $2.3278\times {10}^{-12}$ |

$0.5$ | $1.9519\times {10}^{-5}$ | $1.6110\times {10}^{-7}$ | $6.9550\times {10}^{-10}$ | $0.261168480890467$ | $3.0217\times {10}^{-12}$ |

$0.6$ | $2.4605\times {10}^{-5}$ | $2.0265\times {10}^{-7}$ | $8.7388\times {10}^{-10}$ | $0.383930338842670$ | $3.7949\times {10}^{-12}$ |

$0.7$ | $3.0421\times {10}^{-5}$ | $2.5022\times {10}^{-7}$ | $1.0782\times {10}^{-9}$ | $0.536171515140543$ | $4.6808\times {10}^{-12}$ |

$0.8$ | $3.7293\times {10}^{-5}$ | $3.0647\times {10}^{-7}$ | $1.3199\times {10}^{-9}$ | $0.722781493628416$ | $5.7288\times {10}^{-12}$ |

$0.9$ | $4.5727\times {10}^{-5}$ | $3.7556\times {10}^{-7}$ | $1.6170\times {10}^{-9}$ | $0.950884887178646$ | $7.0170\times {10}^{-12}$ |

$1.0$ | $5.6582\times {10}^{-5}$ | $4.6455\times {10}^{-7}$ | $1.9996\times {10}^{-9}$ | $1.231252940780713$ | $8.6847\times {10}^{-12}$ |

**Table 2.**Comparison of absolute errors in Bessel-QLM for Example 1 using $\beta =\alpha =2$, $M=5,10$, and various $t\in [0,1]$.

t | Bessel-QLM $(\mathit{\alpha}=2)$ | LSM ($\mathit{n}=9$) [23] | TWM [36] | RKM [10] | ||
---|---|---|---|---|---|---|

$\mathit{M}=7$ | $\mathit{M}=10$ | Method (a) | Method (b) | $\mathit{k}=1,{\mathit{M}}_{1}=7$ | $\mathit{n},\mathit{N}=10$ | |

$0.1$ | $4.1979\times {10}^{-8}$ | $2.0195\times {10}^{-11}$ | $1.4169\times {10}^{-5}$ | $1.7830\times {10}^{-7}$ | $2.69611\times {10}^{-5}$ | $1.6674\times {10}^{-5}$ |

$0.2$ | $1.2167\times {10}^{-7}$ | $4.6897\times {10}^{-11}$ | $3.2272\times {10}^{-5}$ | $4.5055\times {10}^{-7}$ | $2.38968\times {10}^{-5}$ | $3.1000\times {10}^{-7}$ |

$0.3$ | $1.8565\times {10}^{-7}$ | $7.4344\times {10}^{-11}$ | $5.1244\times {10}^{-5}$ | $7.1998\times {10}^{-7}$ | $1.01329\times {10}^{-5}$ | $1.1310\times {10}^{-6}$ |

$0.4$ | $2.6108\times {10}^{-7}$ | $1.0203\times {10}^{-10}$ | $7.1441\times {10}^{-5}$ | $1.0081\times {10}^{-6}$ | $2.12408\times {10}^{-5}$ | $2.1200\times {10}^{-6}$ |

$0.5$ | $3.5454\times {10}^{-7}$ | $1.3688\times {10}^{-10}$ | $9.2812\times {10}^{-5}$ | $1.3195\times {10}^{-6}$ | $1.15316\times {10}^{-5}$ | $2.9000\times {10}^{-6}$ |

$0.6$ | $4.1000\times {10}^{-7}$ | $1.5831\times {10}^{-10}$ | $1.1720\times {10}^{-4}$ | $1.6653\times {10}^{-6}$ | $1.85136\times {10}^{-5}$ | $4.1000\times {10}^{-6}$ |

$0.7$ | $5.7919\times {10}^{-7}$ | $2.6467\times {10}^{-10}$ | $1.4496\times {10}^{-4}$ | $2.0620\times {10}^{-6}$ | $1.15473\times {10}^{-5}$ | $6.5000\times {10}^{-6}$ |

$0.8$ | $6.8266\times {10}^{-7}$ | $1.8342\times {10}^{-10}$ | $1.7718\times {10}^{-4}$ | $2.2525\times {10}^{-6}$ | $2.26494\times {10}^{-5}$ | $7.5000\times {10}^{-6}$ |

$0.9$ | $3.0427\times {10}^{-7}$ | $6.5824\times {10}^{-09}$ | $2.1791\times {10}^{-4}$ | $3.1212\times {10}^{-6}$ | $1.13933\times {10}^{-5}$ | $3.3500\times {10}^{-5}$ |

$1.0$ | $3.2344\times {10}^{-5}$ | $3.9874\times {10}^{-07}$ | $2.6999\times {10}^{-4}$ | $3.6311\times {10}^{-6}$ | $8.55545\times {10}^{-5}$ | $4.3700\times {10}^{-5}$ |

**Table 3.**The comparison of numerical results using Bessel-QLM in Example 1 for $M=10$ and various $\alpha $ equal to $\beta =1.5,1.6,\dots ,1.9$.

t | Bessel-QLM $(\mathit{\alpha}=\mathit{\beta})$ | RKM ($\mathit{N}=10$) [10] | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{\beta}=\mathbf{1.5}$ | $\mathbf{\beta}=\mathbf{1.6}$ | $\mathbf{\beta}=\mathbf{1.7}$ | $\mathbf{\beta}=\mathbf{1.8}$ | $\mathbf{\beta}=\mathbf{1.9}$ | $\mathbf{\beta}=\mathbf{1.5}$ | $\mathbf{\beta}=\mathbf{1.6}$ | $\mathbf{\beta}=\mathbf{1.7}$ | $\mathbf{\beta}=\mathbf{1.8}$ | $\mathbf{\beta}=\mathbf{1.9}$ | |

$0.1$ | $0.048353$ | $0.035473$ | $0.025993$ | $0.018983$ | $0.013814$ | $0.047581$ | $0.035141$ | $0.025834$ | $0.018907$ | $0.013779$ |

$0.2$ | $0.140366$ | $0.109646$ | $0.085643$ | $0.066769$ | $0.051926$ | $0.14073$ | $0.10984$ | $0.085716$ | $0.066792$ | $0.051932$ |

$0.3$ | $0.267489$ | $0.215666$ | $0.174210$ | $0.140695$ | $0.113494$ | $0.26686$ | $0.21547$ | $0.17414$ | $0.14067$ | $0.11349$ |

$0.4$ | $0.431675$ | $0.354277$ | $0.292046$ | $0.241173$ | $0.199225$ | $0.43128$ | $0.35420$ | $0.29201$ | $0.24116$ | $0.19922$ |

$0.5$ | $0.639964$ | $0.529749$ | $0.441962$ | $0.370277$ | $0.310871$ | $0.63930$ | $0.52957$ | $0.44188$ | $0.37024$ | $0.31086$ |

$0.6$ | $0.905877$ | $0.750259$ | $0.629314$ | $0.531789$ | $0.451331$ | $0.90494$ | $0.75001$ | $0.62921$ | $0.53175$ | $0.45132$ |

$0.7$ | $1.254902$ | $1.030118$ | $0.863049$ | $0.731769$ | $0.625010$ | $1.2533$ | $1.0297$ | $0.86289$ | $0.73170$ | $0.62499$ |

$0.8$ | $1.740090$ | $1.395364$ | $1.158156$ | $0.979794$ | $0.838525$ | $1.7367$ | $1.3946$ | $1.1579$ | $0.97970$ | $0.83849$ |

$0.9$ | $2.503677$ | $1.899300$ | $1.541363$ | $1.291531$ | $1.102046$ | $2.4864$ | $1.8967$ | $1.5407$ | $1.2913$ | $1.1020$ |

$1.0$ | $4.084549$ | $2.674636$ | $2.066148$ | $1.694499$ | $1.431961$ | $4.0331$ | $2.6677$ | $2.0646$ | $1.6941$ | $1.4318$ |

**Table 4.**The comparison of the numerical results using Bessel-QLM in Example 1 for $M=10$ and various $\alpha $ equal to $\beta =1.5,1.6,\dots ,1.9$ at $t=0.1$.

$\mathit{\beta}$ | Bessel-QLM $(\mathit{\alpha}=\mathit{\beta})$ | RKM ($\mathit{N}=10$) [10] | FDTM ($\mathit{N}=5$) [20] |
---|---|---|---|

$1.9$ | $0.013814$ | $1.3082\times {10}^{-2}$ | $1.3814\times {10}^{-2}$ |

$1.8$ | $0.018983$ | $1.7221\times {10}^{-2}$ | $1.8983\times {10}^{-2}$ |

$1.7$ | $0.025993$ | $2.2504\times {10}^{-2}$ | $2.5993\times {10}^{-2}$ |

$1.6$ | $0.035473$ | $2.9221\times {10}^{-2}$ | $3.5472\times {10}^{-2}$ |

$1.5$ | $0.048353$ | $3.7921\times {10}^{-2}$ | $4.8260\times {10}^{-2}$ |

**Table 5.**The comparison of the numerical results using Bessel-QLM in Example 2 with $M=15,30$, $s=5$, and $\beta =2,\alpha =1$, $\lambda =-{\pi}^{2}$. Numbers in bold show that the correct digits are obtained by the Bessel-QLM.

t | Bessel-QLM | TWM [36] | |||
---|---|---|---|---|---|

${\mathit{U}}_{\mathbf{15},\mathbf{1}}^{\left(\mathbf{6}\right)}\left(\mathit{t}\right)$ | ${\mathcal{E}}_{\mathbf{15},\mathbf{1}}^{\left(\mathbf{6}\right)}\left(\mathit{t}\right)$ | ${\mathit{U}}_{\mathbf{30},\mathbf{1}}^{\left(\mathbf{6}\right)}\left(\mathit{t}\right)$ | ${\mathcal{E}}_{\mathbf{30},\mathbf{1}}^{\left(\mathbf{6}\right)}\left(\mathit{t}\right)$ | $\mathit{K}=\mathbf{1},$${\mathit{M}}_{\mathbf{1}}=\mathbf{11}$ | |

$0.1$ | $-\mathbf{0.269276}38478$ | $8.4777\times {10}^{-8}$ | $-\mathbf{0.26927646955}8501$ | $7.6043\times {10}^{-13}$ | $5.5884\times {10}^{-8}$ |

$0.2$ | $-\mathbf{0.462340}04975$ | $7.2374\times {10}^{-8}$ | $-\mathbf{0.462340122126}269$ | $2.0527\times {10}^{-13}$ | $7.4576\times {10}^{-8}$ |

$0.3$ | $-\mathbf{0.592783}53642$ | $6.4300\times {10}^{-8}$ | $-\mathbf{0.59278360071}7044$ | $3.3595\times {10}^{-13}$ | $1.3357\times {10}^{-8}$ |

$0.4$ | $-\mathbf{0.66837}096929$ | $5.9790\times {10}^{-8}$ | $-\mathbf{0.66837102908}2459$ | $8.9516\times {10}^{-13}$ | $8.9851\times {10}^{-8}$ |

$0.5$ | $-\mathbf{0.6931471}2222$ | $5.8338\times {10}^{-8}$ | $-\mathbf{0.6931471805}61445$ | $1.4998\times {10}^{-12}$ | $1.4347\times {10}^{-7}$ |

$0.6$ | $-\mathbf{0.66837}096929$ | $5.9790\times {10}^{-8}$ | $-\mathbf{0.66837102908}3743$ | $2.1792\times {10}^{-12}$ | $8.9851\times {10}^{-8}$ |

$0.7$ | $-\mathbf{0.592783}53642$ | $6.4300\times {10}^{-8}$ | $-\mathbf{0.59278360071}9678$ | $2.9697\times {10}^{-12}$ | $1.3357\times {10}^{-8}$ |

$0.8$ | $-\mathbf{0.462340}04975$ | $7.2374\times {10}^{-8}$ | $-\mathbf{0.4623401221}30420$ | $3.9452\times {10}^{-12}$ | $7.4576\times {10}^{-8}$ |

$0.9$ | $-\mathbf{0.269276}384782$ | $8.4777\times {10}^{-8}$ | $-\mathbf{0.2692764695}64491$ | $5.2293\times {10}^{-12}$ | $5.5884\times {10}^{-8}$ |

**Table 6.**Numerical solutions in Bessel-QLM for $\beta =1.9,1.7,1.5$ in Example 2 for $M=15$, $\lambda =-{\pi}^{2}$, and $\alpha =1$.

t | $\mathit{\beta}=1.5$ | $\mathit{\beta}=1.7$ | $\mathit{\beta}=1.9$ |
---|---|---|---|

$0.1$ | $-0.294195$ | $-0.294574$ | $-0.279940$ |

$0.2$ | $-0.450191$ | $-0.475380$ | $-0.471703$ |

$0.3$ | $-0.545938$ | $-0.589367$ | $-0.598037$ |

$0.4$ | $-0.599428$ | $-0.652366$ | $-0.669690$ |

$0.5$ | $-0.616646$ | $-0.670828$ | $-0.691846$ |

$0.6$ | $-0.597974$ | $-0.646235$ | $-0.666096$ |

$0.7$ | $-0.539612$ | $-0.576307$ | $-0.591041$ |

$0.8$ | $-0.433043$ | $-0.454769$ | $-0.462076$ |

$0.9$ | $-0.262495$ | $-0.269766$ | $-0.270311$ |

**Table 7.**The comparison of the numerical results using Bessel-QLM in Example 2 with $M=20$, $\beta =1.8$, $\lambda =1,2,3$, and $\alpha =1$.

t | Bessel-QLM | L-RKM [21] | ||||
---|---|---|---|---|---|---|

$\mathbf{\lambda}=\mathbf{1}$ | $\mathbf{\lambda}=\mathbf{2}$ | $\mathbf{\lambda}=\mathbf{3}$ | $\mathbf{\lambda}=\mathbf{1}$ | $\mathbf{\lambda}=\mathbf{2}$ | $\mathbf{\lambda}=\mathbf{3}$ | |

$0.1$ | $0.056413$ | $0.131924$ | $0.261504$ | $0.056405$ | $0.131612$ | $0.260654$ |

$0.2$ | $0.097782$ | $0.231011$ | $0.465679$ | $0.097789$ | $0.230471$ | $0.464144$ |

$0.3$ | $0.125742$ | $0.298753$ | $0.608060$ | $0.125778$ | $0.298048$ | $0.606020$ |

$0.4$ | $0.141008$ | $0.335519$ | $0.684421$ | $0.141075$ | $0.334699$ | $0.682068$ |

$0.5$ | $0.144146$ | $0.342145$ | $0.694381$ | $0.144246$ | $0.341257$ | $0.691916$ |

$0.6$ | $0.135734$ | $0.320218$ | $0.642257$ | $0.135863$ | $0.319304$ | $0.639866$ |

$0.7$ | $0.116398$ | $0.272047$ | $0.536290$ | $0.116553$ | $0.271144$ | $0.534126$ |

$0.8$ | $0.086829$ | $0.200491$ | $0.386913$ | $0.087005$ | $0.199631$ | $0.385081$ |

$0.9$ | $0.047770$ | $0.108725$ | $0.204897$ | $0.047961$ | $0.107936$ | $0.203461$ |

**Table 8.**The comparison of various numerical results with Bessel-QLM for Example 3 for $M=9$, $\beta =2$, $\eta =+1$, and $\alpha =1,2$.

t | Bessel-QLM ($\mathit{\eta}=+1$) | ADM [9] | LOMMs ($\mathit{M}=8$) [34] | ||
---|---|---|---|---|---|

$\mathit{\alpha}=1$ | $\mathit{\alpha}=2$ | Scheme-I | Scheme-II | ||

$0.0$ | $0.00000000000$ | $0.00000000000$ | $0.00000000000$ | $-2.0346\times {10}^{-10}$ | $-5.194068\times {10}^{-8}$ |

$0.1$ | $-0.0016658338$ | $-0.0016658339$ | $-0.0016658339$ | $-0.0016658339$ | $-0.00166586155$ |

$0.2$ | $-0.0066533670$ | $-0.0066533671$ | $-0.0066533671$ | $-0.0066533671$ | $-0.00665336928$ |

$0.5$ | $-0.0411539571$ | $-0.0411539573$ | $-0.0411539568$ | $-0.0411539978$ | $-0.041154078892$ |

$1.0$ | $-0.1588276774$ | $-0.1588276775$ | $-0.1588273537$ | $-0.1588370919$ | $-0.15883515641$ |

**Table 9.**The comparison of (residual) error functions using Bessel-QLM in Example 3 with $M=12$, $\beta =2$ and $\alpha =1,2$.

t | Bessel-QLM ($\mathit{\eta}=+1$) | LOMMs [34] | Bessel-QLM ($\mathit{\eta}=-1$) | JOMMs [37] | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}=1$ | $\mathit{\alpha}=2$ | Scheme-I | Scheme-II | $\mathit{\alpha}=1$ | $\mathit{\alpha}=2$ | $\mathit{\eta}=+1$ | $\mathit{\eta}=-1$ | |

$0.0$ | $0.0000\times {10}^{+00}$ | $0.0000\times {10}^{+00}$ | $2.04\times {10}^{-10}$ | $5.19\times {10}^{-8}$ | $0.0000\times {10}^{+00}$ | $0.0000\times {10}^{+00}$ | − | − |

$0.1$ | $2.7488\times {10}^{-13}$ | $3.0158\times {10}^{-22}$ | $1.88\times {10}^{-11}$ | $5.49\times {10}^{-9}$ | $6.8148\times {10}^{-13}$ | $5.6265\times {10}^{-16}$ | $1.98\times {10}^{-12}$ | $4.66\times {10}^{-11}$ |

$0.2$ | $1.2099\times {10}^{-13}$ | $2.8196\times {10}^{-20}$ | $2.48\times {10}^{-11}$ | $1.73\times {10}^{-8}$ | $4.4922\times {10}^{-13}$ | $1.1621\times {10}^{-13}$ | $1.66\times {10}^{-12}$ | $4.02\times {10}^{-11}$ |

$0.5$ | $4.4965\times {10}^{-15}$ | $6.7454\times {10}^{-19}$ | $8.70\times {10}^{-8}$ | $1.00\times {10}^{-8}$ | $2.9402\times {10}^{-9}$ | $2.9388\times {10}^{-9}$ | $4.64\times {10}^{-10}$ | $5.21\times {10}^{-10}$ |

$1.0$ | $3.6953\times {10}^{-10}$ | $9.4919\times {10}^{-14}$ | $2.53\times {10}^{-5}$ | $2.72\times {10}^{-5}$ | $7.4859\times {10}^{-6}$ | $7.4850\times {10}^{-6}$ | $3.23\times {10}^{-7}$ | $7.63\times {10}^{-7}$ |

**Table 10.**Comparison of numerical results in Bessel-QLM for $M=10$ and $\beta ,\alpha =1.5,1.7,19$ in Example 3.

t | $\mathit{\eta}$$=+1$ | $\mathit{\eta}$$=-1$ | |||||||
---|---|---|---|---|---|---|---|---|---|

Bessel-QLM | HPMADM [25] | Bessel-QLM | |||||||

$\mathbf{\beta}=\mathbf{1.5}$ | $\mathbf{\beta}=\mathbf{1.7}$ | $\mathbf{\beta}=\mathbf{1.9}$ | $\mathbf{\beta}=\mathbf{1.5}$ | $\mathbf{\beta}=\mathbf{1.7}$ | $\mathbf{\beta}=\mathbf{1.9}$ | $\mathbf{\beta}=\mathbf{1.5}$ | $\mathbf{\beta}=\mathbf{1.7}$ | $\mathbf{\beta}=\mathbf{1.9}$ | |

$0.1$ | $-0.0072824$ | $-0.0040292$ | $-0.0022355$ | $-0.0072824$ | $-0.0040292$ | $-0.0022355$ | $-0.0073264$ | $-0.0040412$ | $-0.0022388$ |

$0.2$ | $-0.0204858$ | $-0.0130471$ | $-0.0083268$ | $-0.0204861$ | $-0.0130471$ | $-0.0083268$ | $-0.0208374$ | $-0.0131743$ | $-0.0083720$ |

$0.3$ | $-0.0373729$ | $-0.0258703$ | $-0.0179349$ | $-0.0373745$ | $-0.0258707$ | $-0.0179350$ | $-0.0385599$ | $-0.0263753$ | $-0.0181458$ |

$0.4$ | $-0.0570711$ | $-0.0419374$ | $-0.0308502$ | $-0.0570774$ | $-0.0419393$ | $-0.0308506$ | $-0.0598865$ | $-0.0432809$ | $-0.0314797$ |

$0.5$ | $-0.0790339$ | $-0.0608534$ | $-0.0468938$ | $-0.0790524$ | $-0.0608596$ | $-0.0468956$ | $-0.0845384$ | $-0.0637239$ | $-0.0483641$ |

$0.6$ | $-0.1028640$ | $-0.0823017$ | $-0.0658965$ | $-0.1029100$ | $-0.0823186$ | $-0.0659020$ | $-0.1123910$ | $-0.0876416$ | $-0.0688375$ |

$0.7$ | $-0.1282516$ | $-0.1060115$ | $-0.0876918$ | $-0.1283516$ | $-0.1060515$ | $-0.0877059$ | $-0.1434138$ | $-0.1150420$ | $-0.0929785$ |

$0.8$ | $-0.1549433$ | $-0.1317434$ | $-0.1121130$ | $-0.1551409$ | $-0.1318286$ | $-0.1121454$ | $-0.1776437$ | $-0.1459885$ | $-0.1209032$ |

$0.9$ | $-0.1827273$ | $-0.1592811$ | $-0.1389929$ | $-0.1830894$ | $-0.1594484$ | $-0.1390612$ | $-0.2151732$ | $-0.1805938$ | $-0.1527654$ |

$1.0$ | $-0.2114227$ | $-0.1884270$ | $-0.1681640$ | $-0.2120480$ | $-0.1887346$ | $-0.1682981$ | $-0.2561452$ | $-0.2190184$ | $-0.1887588$ |

**Table 11.**The comparison of numerical solutions using Bessel-QLM with $M=10,15$, $s=5$, and $\beta =2,\alpha =1$ in Example 3. Numbers in bold show that the correct digits are obtained by the Bessel-QLM.

t | Bessel-QLM | LWOMM [36] | |||
---|---|---|---|---|---|

${\mathit{U}}_{\mathbf{10},\mathbf{1}}^{\left(\mathbf{6}\right)}\left(\mathit{t}\right)$ | ${\mathcal{E}}_{\mathbf{10},\mathbf{1}}^{\left(\mathbf{6}\right)}\left(\mathit{t}\right)$ | ${\mathit{U}}_{\mathbf{15},\mathbf{1}}^{\left(\mathbf{6}\right)}\left(\mathit{t}\right)$ | ${\mathcal{E}}_{\mathbf{15},\mathbf{1}}^{\left(\mathbf{6}\right)}\left(\mathit{t}\right)$ | $\mathit{K}=\mathbf{3},$${\mathit{M}}_{\mathbf{1}}=\mathbf{7}$ | |

$0.1$ | $\mathbf{0.3132658}39363106$ | $1.1135\times {10}^{-8}$ | $\mathbf{0.31326585049}5719$ | $2.3444\times {10}^{-12}$ | $1.12567\times {10}^{-11}$ |

$0.2$ | $\mathbf{0.3030154}14602317$ | $8.2300\times {10}^{-9}$ | $\mathbf{0.30301542283}0577$ | $1.7229\times {10}^{-12}$ | $1.06202\times {10}^{-11}$ |

$0.3$ | $\mathbf{0.2860472}58937613$ | $6.3672\times {10}^{-9}$ | $\mathbf{0.28604726530}3516$ | $1.3380\times {10}^{-12}$ | $9.71828\times {10}^{-12}$ |

$0.4$ | $\mathbf{0.26253112}2482223$ | $4.9738\times {10}^{-9}$ | $\mathbf{0.26253112745}4984$ | $1.0495\times {10}^{-12}$ | $8.57608\times {10}^{-12}$ |

$0.5$ | $\mathbf{0.23269678}0038309$ | $3.8355\times {10}^{-9}$ | $\mathbf{0.232696783873}021$ | $8.1360\times {10}^{-13}$ | $7.28159\times {10}^{-12}$ |

$0.6$ | $\mathbf{0.19682680}2831837$ | $2.8611\times {10}^{-9}$ | $\mathbf{0.196826805692}342$ | $6.1151\times {10}^{-13}$ | $5.78329\times {10}^{-12}$ |

$0.7$ | $\mathbf{0.15524810}4677472$ | $2.0053\times {10}^{-9}$ | $\mathbf{0.155248106682}323$ | $4.3376\times {10}^{-13}$ | $4.31280\times {10}^{-12}$ |

$0.8$ | $\mathbf{0.10832276}2202615$ | $1.2419\times {10}^{-9}$ | $\mathbf{0.108322763444}190$ | $2.7506\times {10}^{-13}$ | $2.82023\times {10}^{-12}$ |

$0.9$ | $\mathbf{0.05643860}1915112$ | $5.5412\times {10}^{-10}$ | $\mathbf{0.056438602469}104$ | $1.3218\times {10}^{-13}$ | $1.33990\times {10}^{-12}$ |

**Table 12.**Numerical solutions in Bessel-QLM for $\beta =1.9,1.7,1.5$ in Example 4 for $M=10$ and $\alpha =\beta $.

t | $\mathit{\beta},\mathit{\alpha}=1.5$ | $\mathit{\beta},\mathit{\alpha}=1.7$ | $\mathit{\beta},\mathit{\alpha}=1.9$ |
---|---|---|---|

$0.1$ | $0.4538007$ | $0.3940449$ | $0.3389203$ |

$0.2$ | $0.4218743$ | $0.3736066$ | $0.3259879$ |

$0.3$ | $0.3816409$ | $0.3447748$ | $0.3056717$ |

$0.4$ | $0.3355231$ | $0.3090216$ | $0.2785187$ |

$0.5$ | $0.2850841$ | $0.2674419$ | $0.2450333$ |

$0.6$ | $0.2314604$ | $0.2209409$ | $0.2057129$ |

$0.7$ | $0.1755228$ | $0.1702998$ | $0.1610556$ |

$0.8$ | $0.1179554$ | $0.1162043$ | $0.1115587$ |

$0.9$ | $0.0593026$ | $0.0592595$ | $0.0577135$ |

**Table 13.**The weighted ${L}_{2}$ error norms and the related EOCs in Bessel-QLM for Examples 1–4 for $\beta =2$, $\alpha =1$, and diverse M.

M | Example 1 | Example 2 | Example 3 | Example 4 | ||||
---|---|---|---|---|---|---|---|---|

${\mathcal{L}}_{\mathit{M},\mathit{\alpha}}$ | EOC | ${\mathcal{L}}_{\mathit{M},\mathit{\alpha}}$ | EOC | ${\mathcal{L}}_{\mathit{M},\mathit{\alpha}}$ | EOC | ${\mathcal{L}}_{\mathit{M},\mathit{\alpha}}$ | EOC | |

1 | $1.5958\times {10}^{-1}$ | − | $8.5933\times {10}^{-2}$ | − | $8.5933\times {10}^{-1}$ | − | $1.1185\times {10}^{-1}$ | − |

2 | $3.5672\times {10}^{-2}$ | $2.16$ | $7.9098\times {10}^{-3}$ | $3.44$ | $7.9098\times {10}^{-3}$ | $4.14$ | $1.5476\times {10}^{-2}$ | $2.85$ |

4 | $6.3893\times {10}^{-3}$ | $2.48$ | $7.2809\times {10}^{-4}$ | $3.44$ | $7.2809\times {10}^{-4}$ | $5.94$ | $4.4277\times {10}^{-4}$ | $5.13$ |

8 | $1.3504\times {10}^{-4}$ | $5.56$ | $9.6224\times {10}^{-6}$ | $6.24$ | $9.6224\times {10}^{-8}$ | $12.79$ | $1.4858\times {10}^{-7}$ | $11.54$ |

16 | $4.5362\times {10}^{-8}$ | $11.54$ | $2.8868\times {10}^{-8}$ | $8.38$ | $2.8868\times {10}^{-11}$ | $8.10$ | $6.1046\times {10}^{-11}$ | $11.25$ |

**Table 14.**Residual errors in Bessel-QLM for $\beta =1.1,1.3,1.5,1.7,1.9$ in Example 5 for $M=10$ and $\alpha =\beta $.

t | $\mathit{\beta},\mathit{\alpha}=1.1$ | $\mathit{\beta},\mathit{\alpha}=1.3$ | $\mathit{\beta},\mathit{\alpha}=1.5$ | $\mathit{\beta},\mathit{\alpha}=1.7$ | $\mathit{\beta},\mathit{\alpha}=1.9$ | $\mathit{\beta},\mathit{\alpha}=2.0$ |
---|---|---|---|---|---|---|

$0.1$ | $1.0364\times {10}^{-12}$ | $2.0468\times {10}^{-14}$ | $3.6172\times {10}^{-16}$ | $4.6938\times {10}^{-17}$ | $3.4314\times {10}^{-17}$ | $2.9295\times {10}^{-19}$ |

$0.2$ | $1.0825\times {10}^{-11}$ | $4.0085\times {10}^{-13}$ | $1.1342\times {10}^{-14}$ | $4.7141\times {10}^{-16}$ | $4.4964\times {10}^{-17}$ | $5.6627\times {10}^{-21}$ |

$0.3$ | $4.2966\times {10}^{-11}$ | $2.2006\times {10}^{-12}$ | $8.6122\times {10}^{-14}$ | $4.7959\times {10}^{-15}$ | $8.1274\times {10}^{-16}$ | $6.2154\times {10}^{-20}$ |

$0.4$ | $1.1427\times {10}^{-10}$ | $7.3667\times {10}^{-12}$ | $3.6292\times {10}^{-13}$ | $2.5459\times {10}^{-14}$ | $5.4271\times {10}^{-15}$ | $2.0139\times {10}^{-19}$ |

$0.5$ | $2.4401\times {10}^{-10}$ | $1.8806\times {10}^{-11}$ | $1.1075\times {10}^{-12}$ | $9.2932\times {10}^{-14}$ | $2.3579\times {10}^{-14}$ | $8.3624\times {10}^{-20}$ |

$0.6$ | $4.5356\times {10}^{-10}$ | $4.0444\times {10}^{-11}$ | $2.7559\times {10}^{-12}$ | $2.6760\times {10}^{-13}$ | $7.8496\times {10}^{-14}$ | $1.2112\times {10}^{-19}$ |

$0.7$ | $7.6604\times {10}^{-10}$ | $7.7274\times {10}^{-11}$ | $5.9566\times {10}^{-12}$ | $6.5436\times {10}^{-13}$ | $2.1732\times {10}^{-13}$ | $4.7674\times {10}^{-19}$ |

$0.8$ | $1.2062\times {10}^{-9}$ | $1.3539\times {10}^{-10}$ | $1.1613\times {10}^{-11}$ | $1.4194\times {10}^{-12}$ | $5.2448\times {10}^{-13}$ | $1.2044\times {10}^{-18}$ |

$0.9$ | $1.8003\times {10}^{-9}$ | $2.2204\times {10}^{-10}$ | $2.0928\times {10}^{-11}$ | $2.8112\times {10}^{-12}$ | $1.1418\times {10}^{-12}$ | $8.8587\times {10}^{-18}$ |

**Table 15.**The weighted ${L}_{2}$ error norms and the related EOCs in Bessel-QLM for Example 5 for $\beta ,\alpha =1.01,1.25,1.50,1.75,1.99$, and diverse M.

M | $\mathit{\beta},\mathit{\alpha}=1.01$ | $\mathit{\beta},\mathit{\alpha}=1.25$ | $\mathit{\beta},\mathit{\alpha}=1.50$ | $\mathit{\beta},\mathit{\alpha}=1.75$ | $\mathit{\beta},\mathit{\alpha}=1.99$ | |||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathcal{L}}_{\mathit{M},\mathit{\alpha}}$ | EOC | ${\mathcal{L}}_{\mathit{M},\mathit{\alpha}}$ | EOC | ${\mathcal{L}}_{\mathit{M},\mathit{\alpha}}$ | EOC | ${\mathcal{L}}_{\mathit{M},\mathit{\alpha}}$ | EOC | ${\mathcal{L}}_{\mathit{M},\mathit{\alpha}}$ | EOC | |

1 | $1.9232\times {10}^{-1}$ | − | $1.6545\times {10}^{-1}$ | − | $1.4671\times {10}^{-1}$ | − | $1.3314\times {10}^{-1}$ | − | $1.2312\times {10}^{-1}$ | − |

2 | $1.5972\times {10}^{-2}$ | $3.59$ | $1.4895\times {10}^{-2}$ | $3.47$ | $1.3526\times {10}^{-2}$ | $3.43$ | $1.2065\times {10}^{-2}$ | $3.46$ | $1.0666\times {10}^{-2}$ | $3.53$ |

4 | $2.7591\times {10}^{-5}$ | $9.17$ | $2.0311\times {10}^{-5}$ | $9.52$ | $1.2836\times {10}^{-5}$ | $10.04$ | $6.9934\times {10}^{-6}$ | $10.75$ | $3.3538\times {10}^{-6}$ | $11.64$ |

8 | $4.1872\times {10}^{-12}$ | $22.65$ | $1.2170\times {10}^{-12}$ | $23.99$ | $1.9693\times {10}^{-13}$ | $25.96$ | $1.8278\times {10}^{-14}$ | $28.51$ | $9.7566\times {10}^{-16}$ | $31.68$ |

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