In this research article, we develop a powerful algorithm for numerical solutions to variable-order partial differential equations (PDEs). For the said method, we utilize properties of shifted Legendre polynomials to establish some operational matrices of variable-order differentiation and integration. With the help of the aforementioned operational matrices, we reduce the considered problem to a matrix type equation (equations). The resultant matrix equation is then solved by using computational software like Matlab to get the required numerical solution. Here it should be kept in mind that the proposed algorithm omits discretization and collocation which save much of time and memory. Further the numerical scheme based on operational matrices is one of the important procedure of spectral methods. The mentioned scheme is increasingly used for numerical analysis of various problems of differential as well as integral equations in previous many years. Pertinent examples are given to demonstrate the validity and efficiency of the method. Also some error analysis and comparison with traditional Haar wavelet collocations (HWCs) method is also provided to check the accuracy of the proposed scheme.
Citation: Kamal Shah, Hafsa Naz, Muhammad Sarwar, Thabet Abdeljawad. On spectral numerical method for variable-order partial differential equations[J]. AIMS Mathematics, 2022, 7(6): 10422-10438. doi: 10.3934/math.2022581
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Abstract
In this research article, we develop a powerful algorithm for numerical solutions to variable-order partial differential equations (PDEs). For the said method, we utilize properties of shifted Legendre polynomials to establish some operational matrices of variable-order differentiation and integration. With the help of the aforementioned operational matrices, we reduce the considered problem to a matrix type equation (equations). The resultant matrix equation is then solved by using computational software like Matlab to get the required numerical solution. Here it should be kept in mind that the proposed algorithm omits discretization and collocation which save much of time and memory. Further the numerical scheme based on operational matrices is one of the important procedure of spectral methods. The mentioned scheme is increasingly used for numerical analysis of various problems of differential as well as integral equations in previous many years. Pertinent examples are given to demonstrate the validity and efficiency of the method. Also some error analysis and comparison with traditional Haar wavelet collocations (HWCs) method is also provided to check the accuracy of the proposed scheme.
1.
Introduction
In survey sampling, it is well known fact that suitable use of the auxiliary information may improves the precision of an estimator for the unknown population parameters. The auxiliary information can be used either at the design stage or at estimation stage to increase the accuracy of the population parameter estimators. Several authors presented modified different type of estimators for estimating the finite population mean including [4,9,21,22,23,24,25,26,27].
The problem of estimation of finite population mean or total in two-stage sampling scheme using the auxiliary information has been well established. The two stage sampling scheme is an improvement over the cluster sampling, when it is not possible or easy to calculate all the units from the selected clusters. One of the main characteristic could be the budget, and it becomes too difficult to collect information from all the units within the selected clusters. To overcome this, one way is to select clusters, called first stage unit (fsus) and from the given population of interest, select a subsample from the selected clusters called the second stage units (ssu). This also benefits to increase the size of the first stage samples which consist of clusters, and assume to be heterogeneous groups. If there is no variation within clusters then might not be possible to collect information from all the units within selected clusters. In many situations, it is not possible to obtain the complete list of ultimate sampling units in large scale sample surveys, while a list of primary units of clusters may be available. In such situations, we select a random sample of first stage units or primary units using certain probability sampling schemes i.e simple random sampling (with or without replacement), systematic sampling and probability proportional to size (PPS), and then we can perform sub-sampling in selected clusters (first stage units). This approach is called two-stage sampling scheme.
Two-stage has a great varaity of applications, which go far beyond the immediate scop of sample survey. Whenever any process involves in chemical, physical, or biological tests that can be performed on a small amount of materail, it is likely to be drawn as a subsample from a larger amount that is itself a sample.
In large scale survey sampling, it is usual to adopt multistage sampling to estimate the population mean or total of the study variable y. [13] proposed a general class of estimators of a finite population mean using multi-auxiliary information under two stage sampling scheme. [1] proposed an alternative class of estimators in two stage sampling with two auxiliary variables. [10] proposed estimators for finite population mean under two-stage sampling using multivariate auxiliary information. [12] suggested a detailed note on ratio estimates in multi-stage sampling. [6] given some stratagies in two stage sampling using auxiliary information. [3] suggested a class of predictive estimaotrs in two stage sampling using auxiliary information. [8] gave a generalized method of estimation for two stage sampling using two auxiliary variables. [5] suggested chain ratio estimators in two stage sampling. For certain related work, we refer some latest articles, i.e., [14,15,16,17,18,19,20].
In this article, we propose an improved generalized class of estimators using two auxiliary variables under two-stage sampling scheme. The biases and mean sqaure errors of the proposed generalized class of estimators are derived up to first order of approximation. Based on the numerical results, the proposed class of estimators are more efficient than their existing counterparts.
2.
Symbols and notation
Consider a finite population U = {U1,U2,...,UN} is divided into N first-stage units (fsus) clusters in the population. Let N be the total number of first stage unit in population, n be the number of first stage units selected in the sample, Mi be the number of second stage units (ssus) belongs to the ith first stage units (fsus), (i = 1, 2, …, N), and mi be the number of fsus selected from the ith fsu in the sample of n fsus, (i = 1, 2, …, n).
Let yij, xij and zij be values of the study variable y and the auxiliary variables (xandz) respectively, for the jth ssus Ui=(j=1,2,...,Mi), in the ith fsus. The population mean of the study variable y and the auxiliary variables (x,z) are given by:
In order to obtain the biases and mean sqaured errors, we consider the following relative error terms:
e0=¯y∗−¯Y¯Y,e1=¯x∗−¯X¯X,e2=¯z∗−¯Z¯Z,
E(e20)=λC2by+1nN∑ni=1u2iθiC2iy=Vy,
E(e21)=λC2bx+1nN∑ni=1u2iθiC2ix=Vx,
E(e22)=λC2bz+1nN∑ni=1u2iθiC2iz=Vz,
E(e0e1)=λCbyx+1nN∑ni=1u2iθiCiyx=Vyx,
E(e0e2)=λCbyz+1nN∑ni=1u2iθiCiyz=Vyz,
E(e1e2)=λCbxz+1nN∑ni=1u2iθiCixz=Vxz,
Cby=Sby¯Y,Cbx=Sbx¯X,Cbz=Sbz¯Z,
Cbyx=Sbyx¯Y¯X,Cbyz=Sbyz¯Y¯Z,Cbxz=Sbxz¯X¯Z,
Ciyx=Siyx¯Y¯X,Ciyz=Sbyz¯Y¯Z,Cixz=Sixz¯X¯Z,
Ciy=Siy¯Y,Cix=Six¯X,Ciz=Siz¯Z,
where,
θi=(1mi−1Mi),λ=(1n−1N).
3.
Existing estimators
In this section, we consider several estimators of the finite population mean under two-stage sampling that are available in the sampling literature, the properties of all estimators considered here are obtained up-to the first order of approximation.
(ⅰ) The usual mean estimator ¯y∗=¯y∗0 and its variance under two-stage sampling are given by:
¯y∗0=1n∑ni=1ui¯yi,
(1)
and
V(¯y∗0)=¯Y2Vy=MSE(¯y∗0).
(2)
(ⅱ) The usual ratio estimator under two-stage sampling, is given by:
¯y∗R=¯y∗(¯X¯x∗),
(3)
where ¯X is the known population mean of x.
The bias and MSE of ¯y∗R to first order of approximation, are given by:
Bias(¯y∗R)=¯Y[Vx−Vyx],
(4)
and
MSE(¯y∗R)=¯Y2[Vy+Vx−2Vyx].
(5)
(ⅲ) [2] Exponential ratio type estimator under two-stage sampling, is given by:
¯y∗E=¯y∗exp(¯X−¯x∗¯X+¯x∗).
(6)
The bias and MSE of ¯y∗E to first order of approximation, are given by:
Bias(¯y∗E)=¯Y[38Vx−12Vyx],
(7)
and
MSE(¯y∗E)=¯Y2[Vy+14Vx−Vyx].
(8)
(ⅳ) The traditional difference estimator under two-stage sampling is given by:
¯y∗D=¯y∗+d(¯X−¯x∗),
(9)
where d is the constant.
The minimum variance of ¯y∗D, is given by:
V(¯y∗Dmin)=¯Y2Vy(1−ρ∗2)=MSE(¯y∗D),
(10)
where ρ∗=Vyx√Vy√Vx.
The optimum value of d is dopt=¯YVyx¯XVx.
(ⅴ) [7] Difference type estimator under two-stage sampling, is given by:
¯y∗Rao=d0¯y∗+d1(¯X−¯x∗),
(11)
where d0 and d1 are constants.
The bias and minimum MSE of ¯y∗Rao to first order of approximation, is given by:
The principal advantage of our proposed improved generalized class of estimators under two-stage sampling is that it is more flexible, efficient, than the existing estimators. The mean square errors based on two data sets are minimum and percentage relative efficiency is more than hundred as compared to the existing estimators considered here. We identified 11 estimators as members of the proposed class of estimators by substituting the different values of wi(i=1,2,3), δ and γ. On the lines of [2,7], we propose the following generalized improved class of estimators under two stage sampling for estimation of finite population mean using two auxiliary varaible as given by:
where wi(i=1,2,3) are constants, whose values are to be determined; δ and γ are constants i.e., (0≤δ, γ≤1) and can be used to construct the different estimators.
Using (25), solving ¯y∗G in terms of errors, we have
Solving (27), the minimum MSE of ¯y∗G to first order of approximation are given by:
MSE(¯y∗G)min=¯Y2[1−Ω24Ω1],
(28)
where
Ω1=ABC−AI2−BH2−CG2+2GHI+BC−I2,
and
Ω2=ABF2+ACE2−2AEFI+BCD2−2BDFH−2CDEG−D2I2+2DEHI
+2DFGI−E2H2+2EFGH−F2G2+4BCD+BF2−4BFH+CE2
−4CEG−4DI2−2EFI+4EHI+4FGI+4BC+4I2.
The optimum values of wi(i=1,2,3) are given by:
w1opt=Ω32Ω1,w2opt=¯YΩ42¯XΩ1, and w3opt=¯YΩ52¯ZΩ1,
where
Ω3=BCD−BFH−CEG−DI2+EHI+FGI+2GI+2BC−2I2,
Ω4=ACE−AFI−CDG+DHI−EH2+FGH+CE−2CG−FI+2HI,
Ω5=ABF−AEI−BDH+DGI+EGH−FG2+BF−2BH−EI+2GI.
From (28), we produce the following two estimators called ¯y∗G1 and ¯y∗G2. Put (δ=0,γ=1) and (δ=1,γ=0) in (25), we get the following two estimators respectively:
Population 1. [Source: [11], Model Assisted Survey Sampling]
There are 124 countries (second stage units) divided into 7 continents (first stage units) according to locations. Continent 7th consists of only one country therefore, we placed 7th continent in 6th continent.
We considered:
y = 1983 import (in millions U.S dollars),
x = 1983 export (in millions U.S dollars),
z = 1982 gross national product (in tens of millions of U.S dollars).
The data are divided into 6 clusters, having N=6, and n=3. Also ∑Ni=1Mi=124, ¯M=20.67. In Table 2, we show cluster sizes, and population means of the study variable (y) and the auxiliary variables (x,z). Tables 3 and 4 give some results.
Population 2. [Source: [11], Model Assisted Survey Sampling]
Similarly we considered the data as mentioned in Population 1,
y = 1983 import (in millions U.S dollars),
x = 1981 military expenditure (in tens of millions U.S dollars),
z = 1980 population (in millions).
The data are divided into 6 clusters having N=6, n=3, ∑Ni=1Mi=124,¯M=20.67.
In Table 5, we show cluster sizes, and means of the study variable (y) and the auxiliary variables (x,z). Tables 6 and 7 give some computation results.
The results based on Tables 2–7 are given in Tables 8 and 9 having biasses, mean square errors, and percentage relative efficiencies of the poposed and exisitng estimators w.r.t ¯y∗0.Tables 8 and 9 show that the proposed estimators perform well as compared to the existing estimators considered here.
Table 8.
Biases of different estimators in both data sets.
The following expression is used to obtain the Percent Relative Efficiency (PRE), i.e.,
PRE=MSE(¯y∗0)MSE(¯y∗i)×100,
where i=0,R,E,D,Rao,DR,DE,DD,DD(R),G1,G2.
6.
Discussion
As mentioned above, we used two real data sets to obtain the biases, MSEs or variances and PREs of all estimators under two-stage sampling scheme when using two auxiliary variables. In Tables 2–4 and Tables 5–7, we present the summary statistic of both population. From Tables 8 and 9, we observed that the proposed class of estimators ¯y∗G1 and ¯y∗G2 are more precise than the existing estimators ¯y∗0, ¯y∗R, ¯y∗E, ¯y∗D, ¯y∗Rao, ¯y∗DR, ¯y∗DE, ¯y∗DD, ¯y∗DD(R) in terms of MSEs and PREs. It is clear that the proposed improved generalized class of estimators, i.e., performs better than the estimators. As we increase the sample size the mean square error values decreases, and percentage relative efficiency give best results, which are the expected results.
7.
Conclusions
In this manuscript, we proposed a generalized class of estimators using two auxiliary variables under two-stage sampling for estimating the finite population mean. In addition, some well-known estimators of population mean like traditional unbiased estimator, usual ratio, exponential ratio type, traditional difference type, Rao difference type, difference-in- ratio type, difference-in-exponential ratio type, difference-in-difference, difference-difference ratio type estimator are created to be members of our suggested improved generalized class of estimators. Expression for the biases and mean squared error have been generated up to the first order of approximation. We identified 11 estimators as members of the proposed class of estimators by substituting the different values of wi(i=1,2,3), δ and γ. Both generalized class of estimators ¯y∗G1 and ¯y∗G2 perform better as compared to all other considered estimators, although ¯y∗G2 is the best. In Population 2, the performance of ratio estimator (¯y∗R) is weak. The gain in Population 1 is more as compared to Population 2.
Acknowledgments
The authors are thankful to the Editor-in-Chief and two anonymous referees for their careful reading of the paper and valuable comments which leads to a significant improvement in article.
Conflict of interest
The authors declare no conflict of interest.
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Figure 1. Graphical presentation of approximate solutions and absolute error of Example 1 at α(t)=1+exp(−t), β(t)=1 and Scale level 6
Figure 2. Graphical presentation of approximate solutions and absolute error of Example 1 at α(t)=2, β(t)=1−exp(−t) and Scale level 6
Figure 3. Graphical presentation of approximate solutions and absolute error of Example 2 at various values of α=2−sin(t),β(t)=1−exp(−t), and scale level 6
Figure 4. Graphical presentation of approximate solutions and absolute error of Example 2 at various values of α=t2+1.52,with0<t<≤1,β(t)=1−t22, and scale level 6
Figure 5. Graphical presentation of approximate solutions and absolute error of Example 3 at various values of α=1+exp(−πt),β(t)=1, and scale level 6
Figure 6. Graphical presentation of approximate solutions and absolute error of Example 3 at various values of α=2−cos(t),β(t)=1, and scale level 6