Classes of Ordinary Differential Equations for the Alpha Power Lomax Distribution

: This paper focuses on the ordinary differential equations (ODEs) of the alpha-power Lomax distribution. The ODEs are derived from the alpha-power Lomax (APL) distribution for the density, quantile, survival and hazard functions. These ODEs helps to study the shape of various functions that describe the probability distribution. Under necessary conditions of the parameters, the existence of the ODEs is possible and obtained.


Introduction
Several continuous life time distributions have been used to deal with the reliability modeling of various fields like engineering, medical science, business, finance, insurance, etc.The Lomax distribution is one of them that were originally introduced by Lomax [1] for the modeling the failure data in business and engineering.Harris [2] used the Lomax distribution for income and wealth data.Bryson [3] considered this distribution while modeling the heavy tailed data.Its applications were discussed in detail by Hassan and Ghamdi [4] in reliability modeling and life tests experiments.Besides, due to the widespread applications of the Lomax distribution, it was further modified so as to increase the flexibility of the data.For example, Poisson-Lomax distribution by Al-Awadhi and Ghitany [5], double Lomax distribution by Punathumparamhath [6].For more detail study, we refer to see [7][8][9][10][11][12][13][14][15][16][17][18][19][20].However, these generalizations are not adequate since their flexibilities are limited.
Several mathematical disciplines are significantly utilised probability distributions like Calculus, differential equations, algebra, measure theory, etc. Ordinary differential equations (ODE) have primarily been used up to now for parameter estimation and approximation.Approximation of quantile function features prominently in the use of ODE in approximation [21].The ODEs of the distributions including beta distribution, Lomax distribution, beta prime distribution, Laplace distribution and raised cosine distribution can be found in literature [22].However, the ODEs have not been studied before neither for the alpha-power Lomax distribution or other extensions of the Lomax distribution.Finally, it is noteworthy that spectral element method [23][24][25][26][27] due to its high accuracy as a numerical method can be utilized in modeling of similar problems.
The novelty of this paper to examine some classes of ODE, s for the APL distribution.

Ordinary differential equations of the alpha power lomax distribution
The cumulative distribution function (cdf) of APT is given by ( ( ) the probability density function (pdf) is given as Using the cdf of the Lomax distribution eq (2) yields the pdf of APL distribution

ODEs for the PDF
The first order ODEs Eq. (1) provides ( ) For the solution of Eq (4), it is required a, b, α , x > 0 and α ≠ 1.The simplified form of Eq. ( 4) is ODEs for the PDF with different parameter values ( 1) ( ) 0.6 2( 1) ( 2) ( ) 2.4 3( 2) The ODEs for Eq. ( 5) can be solved only when the parameter values are specified.Table 1 describes a class of ODEs in respect of various parameter values.

ODEs for the quantile function (QF)
The QF of APL distribution is obtained as By differentiating Eq. ( 11), we get the first order ODEs (12) ( ) The solution of Eq. ( 12) is possible when a, b, α, x ˃ 0, α ≠ 1.The simplified form of Eq. ( 11) is given by Putting Eq. ( 13) in Eq. ( 12), we obtain ( ) ( ) Eq. ( 14) can be solved for the unique values of the parameters.Table 2 reflects the ODEs with their specified parameter values.
Differentiate Eq. ( 15) to get the second order derivative Then, we have ( )

ODEs for the survival function (SF)
The SF of APL distribution is given by Eq. ( 17) can be rewritten as Putting Eq. ( 19) in Eq. ( 18), we get The ODEs for Eq. ( 21) can be obtained only when the parameter values are specified.Table 3 reflects the numerical results against specified parameter values.

ODEs for the hazard function (HF)
The HF of APL distribution is defined by By differentiating Eq. ( 24), we get the following first order ODEs Table 4 reflects the ODEs for the HF of APL distribution with the specified parameter values.

Application
The data set defines the mortality rate of COVID-19 in Pakistan which was taken from the URL https://github.com/owid/covid-19-data.

Table 2 .
ODEs for the QF with different parameter values

Table 3 .
ODEs for the SF with different parameter values

Table 4 .
ODEs for the HF with different parameter values