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Numerical Solution of Boundary-Value and Initial-Boundary-Value Problems Using Spline Functions

Thesis Info

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External Link

Program

PhD

Institute

Ghulam Ishaq Khan Institute of Engineering Sciences and Technology

City

Swabi

Province

KPK

Country

Pakistan

Thesis Completing Year

2009

Thesis Completion Status

Completed

Subject

Applied Sciences

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/2882/1/333S.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726779758

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The following two types of problems in differential equations are investigated: (i) Second and sixth-order linear and nonlinear boundary-value problems in ordinary differential equations using non-polynomial spline functions. (ii) One dimensional nonlinear Initial-boundary-value problems in partial differential equations using B-spline collocation method. Polynomial splines, non-polynomial splines and B-splines are introduced. Some well known results and preliminary discussion about convergence analysis of boundary-value problems and stability theory are described. Quartic non-polynomial spline functions are used to develop numerical methods for computing approximations to the solution of linear, nonlinear and system of second- order boundary-value problems and singularly perturbed boundary-value problems. Convergence analysis of the method is discussed. Numerical methods for computing approximations to the solution of linear and nonlinear sixth-order boundary-value problems with two-point boundary conditions are developed using septic non-polynomial splines. Second-, Fourth- and Sixth-order convergence is obtained. Numerical method based on collocation method using quartic B-spline functions for the numerical solution of one-dimensional modified equal width (MEW) wave equation is developed. The scheme is shown to be unconditionally stable using Von-Neumann approach. Propagation of a single wave, interaction of two waves and Maxwellian initial condition are discussed. Algorithms based on quartic and Quintic B-spline collocation methods are designed for the numerical solution of the modified regularized long wave (MRLW) equation. Stability analysis is performed. Propagation of a solitary wave, interaction of multiple solitary waves, and generation of train of solitary waves are also investigated. Quartic and quintic B-spline functions have been used to develop collocation methods for the numerical solution of Kuramoto-Sivashinsky (KS) equation. Also, using splitting technique, the equation is reduced to a problem of second order in space. Using error norms L2 and L∞ and conservative properties of mass, momentum and energy, accuracy and efficiency of the suggested methods is established through comparison with the existing numerical techniques. Performance of the algorithms is tested through application of the methods on benchmark problems.
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۔غزل

غزل---مبشر سعید،رجسٹراریونیورسٹی آف ساہیوال

کوئی ملتا ہی نہیں سوختہ پا میری طرح
میرے جیون کو اُداسی سے ملانے والا!
میں نے احباب کو آواز لگا کر پوچھا
اے کئی دن سے مرے ذہن پہ چھائے ہوئے شخص
رات بھر چاند کو احوال سنانے کے لیے
زندگی! میری طرف دیکھ کے ایماں سے بتا
تند اور تیز ہواؤں کے علاقے میں سعید

 

جس کو معلوم ہو وحشت کا پتا میری طرح
دشت میں پھِرتا رہے آبلہ پا میری طرح
کوئی رہتا ہے شبِ غم میں سدا، میری طرح؟
تو مجھے وصل کے سپنے نہ دکھا میری طرح
کیا ٹھہرتی ہے دریچوں میں ہوا، میری طرح؟
ایک بھی شخص کوئی تجھ کو مِلا، میری طرح
زیست کرتا ہے فقط دل کا دِیا میری طرح

Work Fatigue Determination of Nurses in Hospital of Hasanuddin University

This study aims to determine the effect of body mass index (BMI), sleep patterns, working years, physical workload, mental workload, and work time on work fatigue on nurses at Hasanuddin University Hospital Makassar. This research is a quantitative study with a cross-sectional design with a sample size of 56 people. Sampling using a simple random sampling method. Data collection was carried out by means of a questionnaire. Bodyweight is measured by weight scales, body temperature is measured by microtoice, and physical workload is measured by a pulse oximeter, which is by looking at the pulse of the nurse before and after work. Data analysis in this study used the SPSS program. The results of this study indicate that there is an influence between Body Mass Index (BMI) on work fatigue (p = 0.001), there is an effect of sleep patterns on work fatigue (p = 0.019), there is an effect of tenure on work fatigue (p = 0.017), the relationship between physical workload and work fatigue (p = 0.000), there is an effect of mental workload on work fatigue (p = 0.000) and there is an effect of length of work on work fatigue (p = 0.017). The physical workload is the variable with the greatest influence on work fatigue on nurses at Hasanuddin University Hospital Makassar with Exp. (B) = 14,526. The results of this study concluded that there was a significant influence between the Body Mass Index (BMI), sleep patterns, years of work, physical workload, mental workload, and length of work for nurses at Hasanuddin University Hospital Makassar.

Mean Estimation in Adaptive Cluster Sampling Using Auxiliary Information

In this dissertation, generalized simple and exponential type estimators have been developed using the information of single and two auxiliary variables for the estimation of rare and clustered population mean in adaptive cluster sampling designs. The proposed estimators are specifically developed for different situations of clustered populations in simple adaptive cluster sampling, stratified adaptive cluster sampling and systematic adaptive cluster sampling designs. In Chapter 1, the discussion has been made about the situations of rare and clustered population in which the conventional sampling designs may not be appropriate in order to achieve even moderate precision. The use of adaptive cluster sampling design along with the process in the presence of auxiliary information is also discussed. Comparison of adaptive cluster sampling with conventional sampling design and some advantages and disadvantages has also been given. Furthermore, stratified adaptive cluster sampling and systematic adaptive cluster sampling has been illustrated in the same Chapter with the detail sampling process. In Chapter 2, the literature regarding the use of auxiliary information in conventional sampling designs, adaptive cluster sampling, stratified adaptive cluster sampling and systematic adaptive cluster sampling have been discussed whereas Chapter 3 contains some basic estimators that already developed in conventional sampling designs, adaptive cluster sampling, stratified adaptive cluster sampling and systematic adaptive cluster sampling designs. The major contribution of this dissertation appears from Chapter 4 by proposing modified ratio and regression-cum-modified ratio estimators using the information of single auxiliary variable in adaptive cluster sampling by utilizing the average values of the networks with simple random sampling without replacement. The expressions of approximate bias and mean square error for the proposed estimators have been derived. The generalized form for the proposed estimators has been suggested by introduction the unknown constants. The expressions of approximate bias and mean square error have been derived for the generalized form and optimum properties have been discussed. Many conventional and non-conventional parameters of the auxiliary variable have been used as special cases of the proposed estimators. The efficiency issues in adaptive cluster sampling have also been discussed. Theoretical comparisons have been made of the proposed estimators with existing estimators. An extensive numerical study is conducted by using real and artificial population data sets for all the estimators to evaluate their performance. In Chapter 5, weighted exponential ratio-product type estimator have been developed using single auxiliary variable in adaptive cluster sampling for the situations in which the relationship between the survey variable and the auxiliary variable is non-linear. The expressions of approximate bias and mean square error have been derived. A simulation study is conducted to evaluate the performance of the proposed estimator with existing exponential type estimators. In Chapter 6, a generalized semi-exponential type estimator has been suggested based on two auxiliary variables in adaptive cluster sampling. Some exponential and non-exponential type estimators have been discussed, as the special cases of the proposed estimator. The expressions of estimated bias and minimum mean square error have been derived. A simulation study is conducted on simulated populations generated by Poisson cluster process and Ecodist Package in R, to examining the performance of proposed estimator in adaptive cluster sampling design. In Chapter 7, modified ratio and regression-cum-modified ratio estimators have been developed using the information of single auxiliary variable in stratified adaptive cluster sampling. The generalized form for the proposed estimators has been suggested by introduction the unknown constants. The expressions of approximate bias and mean square error have been derived and optimum properties have been discussed. Theoretical comparisons have been made of the proposed estimators with existing estimators. An extensive numerical study is conducted by using real and artificial population data sets for all the estimators to evaluate their performance. In Chapter 8, a generalized semi-exponential type estimator has been suggested based on two auxiliary variables by utilizing the average values of the networks in stratified adaptive cluster sampling. Some exponential and nonexponential type estimators have been discussed, as the special cases of the proposed estimator. The expressions of approximate bias and minimum mean square error have also been derived. A simulation study is conducted using the simulated populations generated by Poisson cluster process at different level of rarity and aggregation to examining the performance of proposed estimator in stratified adaptive cluster sampling design. In Chapter 9, modified ratio and regression-cum-modified ratio estimators have been developed using the information of single auxiliary variable in systematic adaptive cluster sampling. The generalized form for the proposed estimators has been suggested by introducing the unknown constants. The expressions of approximate bias and mean square error have been derived and optimum properties have been discussed. Theoretical comparison has been made of the proposed estimators with existing estimators. A numerical study is conducted by using artificial population data sets taken from Thompson (2012) for all the estimators to evaluate their performance.