Applied Mathematics

Textbook:Applied Mathematics for Scientists and Engineers
Author(s): Youssef N. Raffoul

Course description:
Ordinary differential equations
• Partial differential equations
• Matrices and systems of linear equations
• Calculus of variations
• Integral equations


Chapter 1 is a review of ordinary differential equations and is not intended to be formally covered by the instructor. It is recommended that students become acquainted with it before proceeding to the following chapters. The main reason for including Chapter 1 is that by the time students take a graduate course in applied mathematics, they have already forgotten most techniques for solving ordinary differential equations. In addition, it will save class time by not formally reviewing such topics but rather asking the students to read them beforehand. The chapter covers first-order and higher-order differential equations. It also includes a section on the Cauchy-Euler equation, which plays a significant role in Chapters 4 and 5.
The second chapter is devoted to the study of partial differential equations, with the majority of the content aimed toward graduate students pursuing engineering degrees. The chapter begins with linear equations with constant and variable coefficients and then moves on to quasi-linear equations. Burger’s equation occupies an important role in the chapter, as do second-order partial differential equations and homogeneous and nonhomogeneous wave equations.
The third chapter discusses matrices and systems of linear equations. Gauss elimination, matrix algebra, vector spaces, and eigenvalues and eigenvectors are all covered. The chapter concludes with an examination of inner product spaces, diagonalization, quadratic forms, and functions of symmetric matrices.
Chapter 4 delves deeply into fundamental themes in the calculus of variations in a functional analytic environment. The calculus of variations is concerned with the optimization of functionals over a set of competing objects. We begin by deriving the Euler-Lagrange necessary condition and generalizing the concept to functionals with higher derivatives or with multiple variables. We provide a nice discussion on the theory behind sufficient conditions. Some of the topics are generalized to isoperimetric problems and functionals with constraints. Toward the end of the chapter, we closely examine the connection between the Sturm-Liouville problem and the calculus of variations. We end the chapter with the Rayleigh-Ritz method and the development of Euler-Lagrange to allow variational computation of multiple integrals.
Chapter 5 is solely devoted to the study of Fredholm and Volterra integral equations. The chapter begins by introducing integral equations and the connections between them and ordinary differential equations. The development of Green’s function occupies an important role in the chapter. It is used to classify kernels, which in turn leads us to the appropriate approach for finding solutions. This includes integral equations with symmetric kernels or degenerate kernels. Toward the end of the chapter, we develop iterative methods and the Neumann series. We briefly discuss ways of approximating non-degenerate kernels and the use of the Laplace transform in solving integral equations of convolution types. Since not all integral equations can be reduced to differential equations, one should expect odd behavior from solutions. For such reasons, we devote the last section of the chapter to the qualitative analysis of solutions using fixed point theory and the Liapunov direct method.