iii ACKNOWLEDGMENTS The writing of this dissertation would not be possible without the support of numerous people. My sincere gratitude goes to my parents who hail from the mountain of Nepal, encourage me for my higher education, and support me in every steps of my academic career. My deepest gratitude goes to my advisor Dr. Jiahong Wu for his continuous and expert guidance throughout my graduate studies. His patience, genuine caring and concern, and availability for all of his students is incredible. I thank him for his motivating, encouraging, and enlightening teaching that guided me to pursue a career in Applied Mathematics. I am forever grateful to him for his faith in me. I am thankful to Dr. Juan-Ming Yuan from Providence University o...