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It is significantly important to search for exact soliton solutions to nonlinear partial differential equations (PDEs) of mathematical physics. Transforming nonlinear PDEs into bilinear forms using the Hirota differential operators enables us to apply the Wronskian and Pfaffian techniques to search for exact solutions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili (KP) equation with not only constant coefficients but also variable coefficients under a certain constraint (ut + α 1(t)uxxy + 3α 2(t)uxuy)x +α 3 (t)uty -α 4(t)uzz + α 5(t)(ux + α 3(t)uy) = 0. However, bilinear equations are the nearest neighbors to linear equations, and expected to have some properties similar to those of linear equations. We have explored a key feature of the linear superposition principle, which linear differential equations have, for Hirota bilinear equations, while intending to construct a particular sub-class of N-soliton solutions formed by linear combinations of exponential traveling waves. Applications are given for the (3+1) dimensional KP, Jimbo-Miwa (JM) and BKP equations, thereby presenting their particular N-wave solutions. An opposite question is also raised and discussed about generating Hirota bilinear equations possessing the indicated N-wave solutions, and two illustrative examples are presented. Using the Pfaffianization procedure, we have extended the generalized KP equation to a generalized KP system of nonlinear PDEs. Wronskian-type Pfaffian and Gramm-type Pfaffian solutions of the resulting Pfaffianized system have been presented. Our results and computations basically depend on Pfaffian identities given by Hirota and Ohta. The Pl̈ucker relation and the Jaccobi identity for determinants have also been employed. A (3+1)-dimensional JM equation has been considered as another important example in soliton theory, uyt - uxxxy - 3(uxuy)x + 3uxz = 0. Three kinds of exact soliton solutions have been given: Wronskian, Grammian and Pfaffian solutions. The Pfaffianization procedure has been used to extend this equation as well. Within Wronskian and Pfaffian formulations, soliton solutions and rational solutions are usually expressed as some kind of logarithmic derivatives of Wronskian and Pfaffian type determinants and the determinants involved are made of functions satisfying linear systems of differential equations. This connection between nonlinear problems and linear ones utilizes linear theories in solving soliton equations. B̈acklund transformations are another powerful approach to exact solutions of nonlinear equations. We have computed different classes of solutions for a (3+1)-dimensional generalized KP equation based on a bilinear B̈acklund transformation consisting of six bilinear equations and containing nine free parameters. A variable coefficient Boussinesq (vcB) model in the long gravity water waves is one of the examples that we are investigating, ut + α 1 (t)uxy + α 2(t)(uw)x + α 3(t)vx = 0; vt + β1(t)(wvx + 2vuy + uvy) + β2(t)(uxwy - (uy)2) + β3(t)vxy + β4(t)uxyy = 0, where wx = uy. Double Wronskian type solutions have been constructed for this (2+1)-dimensional vcB model.

The aim of this work is to use the Pfaffian technique, along with the Hirota bilinear method to construct different classes of exact solutions to various of generalized integrable nonlinear partial differential equations. Solitons are among the most beneficial solutions for science and technology, from ocean waves to transmission of information through optical fibers or energy transport along protein molecules. The existence of multi-solitons, especially three-soliton solutions, is essential for information technology: it makes possible undisturbed simultaneous propagation of many pulses in both directions. The derivation and solutions of integrable nonlinear partial differential equations in two spatial dimensions have been the holy grail in the field of nonlinear science since the late 1960s. The prestigious Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations, as well as the ,Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable nonlinear partial differential equations in (1+1) and (2+1) dimensions, respectively. Do there exist Pfaffian and soliton solutions to generalized integrable nonlinear partial differential equations in (3+1) dimensions? In this dissertation, I obtained a set of explicit exact Wronskian, Grammian, Pfaffian and N-soliton solutions to the (3+1)-dimensional generalized integrable nonlinear partial differential equations, including a generalized KP equation, a generalized B-type KP equation, a generalized modified B-type KP equation, soliton equations of Jimbo-Miwa type, the nonlinear Ma-Fan equation, and the Jimbo-Miwa equation. A set of sufficient conditions consisting of systems of linear partial differential equations involving free parameters and continuous functions is generated to guarantee that the Wronskian determinant or the Pfaffian solves these generalized equations. On the other hand, as part of this dissertation, bilinear Bäcklund transformations are formally derived for the (3+1)-dimensional generalized integrable nonlinear partial differential equations: a generalized B-type KP equation, the nonlinear Ma-Fan equation, and the Jimbo-Miwa equation. As an application of the obtained Bäcklund transformations, a few classes of traveling wave solutions, rational solutions and Pfaffian solutions to the corresponding equations are explicitly computed. Also, as part of this dissertation, I would like to apply the Pfaffianization mechanism of Hirota and Ohta to extend the (3+1)-dimensional variable-coefficient soliton equation of Jimbo-Miwa type to coupled systems of nonlinear soliton equations, called Pfaffianized systems. Examples of the Wronskian, Grammian, Pfaffian and soliton solutions are explicitly computed. The numerical simulations of the obtained solutions are illustrated and plotted for different parameters involved in the solutions.

An average heat transfer coefficient, h_bar, is often used to solve heat transfer problems. It should be understood that this is an approximation and may provide inaccurate results, especially when the temperature field is of interest. The proper method to solve heat transfer problems is with a conjugate approach. However, there seems to be a lack of clear explanations of conjugate heat transfer in literature. The objective of this work is to provide a clear explanation of conjugate heat transfer and to determine the discrepancy in the temperature field when the interface boundary condition is approximated using h_bar compared to a local, or variable, heat transfer coefficient, h(x). Simple one-dimensional problems are presented and solved analytically using both h(x) and h_bar. Due to the one-dimensional assumption, h(x) appears in the governing equation for which the common methods to solve the differential equations with an average coefficient are no longer valid. Two methods, the integral equation and generalized Bessel methods are presented to handle the variable coefficient. The generalized Bessel method has previously only been used with homogeneous governing equations. This work extends the use of the generalized Bessel method to non-homogeneous problems by developing a relation for the Wronskian of the general solution to the generalized Bessel equation. The solution methods are applied to three problems: an external flow past a flat plate, a conjugate interface between two solids and a conjugate interface between a fluid and a solid. The main parameter that is varied is a combination of the Biot number and a geometric aspect ratio, A_1^2 = Bi*L^2/d_1^2. The Biot number is assumed small since the problems are one-dimensional and thus variation in A_1^2 is mostly due to a change in the aspect ratio. A large A_1^2 represents a long and thin solid whereas a small A_1^2 represents a short and thick solid. It is found that a larger A_1^2 leads to less problem conjugation. This means that use of h_bar has a lesser effect on the temperature field for a long and thin solid. Also, use of ¯ over h(x) tends to generally under predict the solid temperature. In addition is was found that A_2^2, the A^2 value for the second subdomain, tends to have more effect on the shape of the temperature profile of solid 1 and A_1^2 has a greater effect on the magnitude of the difference in temperature profiles between the use of h(x) and h_bar. In general increasing the A^2 values reduced conjugation.