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ABSTRACTThis book is a collection of papers in memory of Gu Chaohao on the subjects of Differential Geometry, Partial Differential Equations and Mathematical Physics that Gu Chaohao made great contributions to with all his intelligence during his lifetime. All contributors to this book are close friends, colleagues and students of Gu Chaohao. They are all excellent experts among whom there are 9 members of the Chinese Academy of Sciences. Therefore this book will provide some important information on the frontiers of the related subjects. "

"This book is a collection of papers in memory of Gu Chaohao on the subjects of Differential Geometry, Partial Differential Equations and Mathematical Physics that Gu Chaohao made great contributions to with all his intelligence during his lifetime. All contributors to this book are close friends, colleagues and students of Gu Chaohao. They are all excellent experts among whom there are 9 members of the Chinese Academy of Sciences. Therefore this book will provide some important information on the frontiers of the related subjects."

"This book is designed to serve as a textbook for a course on ordinary differential equations, which is usually a required course in most science and engineering disciplines and follows calculus courses. The book begins with linear algebra, including a number of physical applications, and goes on to discuss first-order differential equations, linear systems of differential equations, higher order differential equations, Laplace transforms, nonlinear systems of differential equations, and numerical methods used in solving differential equations. The style of presentation of the book ensures that the student with a minimum of assistance may apply the theorems and proofs presented. Liberal use of examples and homework problems aids the student in the study of the topics presented and applying them to numerous applications in the real scientific world. This textbook focuses on the actual solution of ordinary differential equations preparing the student to solve ordinary differential equations when exposed to such equations in subsequent courses in engineering or pure science programs. The book can be used as a text in a one-semester core course on differential equations, alternatively it can also be used as a partial or supplementary text in intensive courses that cover multiple topics including differential equations."

"This book offers engineering students an introduction to the theory of partial differential equations and then guiding them through the modern problems in this subject.Divided into two parts, in the first part readers already well-acquainted with problems from the theory of differential and integral equations gain insights into the classical notions and problems, including differential operators, characteristic surfaces, Levi functions, Green’s function, and Green’s formulas. Readers are also instructed in the extended potential theory in its three forms: the volume potential, the surface single-layer potential and the surface double-layer potential. Furthermore, the book presents the main initial boundary value problems associated with elliptic, parabolic and hyperbolic equations. The second part of the book, which is addressed first and foremost to those who are already acquainted with the notions and the results from the first part, introduces readers to modern aspects of the theory of partial differential equations. "

"The book covers several topics of current interest in the field of nonlinear partial differential equations and their applications to the physics of continuous media and particle interactions. It treats the quasigeostrophic equation, integral diffusions, periodic Lorentz gas, Boltzmann equation, and critical dispersive nonlinear Schrödinger and wave equations. The book describes in a careful and expository manner several powerful methods from recent top research articles."

"Model pertumbuhan populasi dalam persamaan diferensial parsial (PDP) menggambarkan evolusi jumlah populasi dalam spasial dan waktu. Dalam penerapannya, telah diaplikasikan model PDP dalam ilmu matematika biologi yang disebut Model Diffusive Malthus dan Model Fisher-Kolmogorov.  Pada skripsi ini, model tersebut dikaji kembali dan dimodifikasi menjadi Model Modifikasi Fisher-Kolmogorov, di mana termasuk ke dalam persamaan reaksi-difusi yang dibentuk dari persamaan difusi dan persamaan logistik dengan melibatkan efek perburuan sebagai suku reaksinya. Kedua suku tersebut dan masing-masing parameter di dalamnya memiliki peranan penting karena digunakan untuk mempelajari perilaku solusi, baik secara analitik maupun numerik. Analisis numerik serta simulasinya untuk solusi model ini dikerjakan menggunakan metode beda hingga eksplisit berdasarkan kondisi nilai awal.

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Population growth model on partial differential equation (PDE) describes the evolution of the number of population in spatial and time. In this application, there has been applied the PDE model in mathematical biology that is called Diffusive Malthus Model and Fisher-Kolmogorov Model. In this thesis, those model are reviewed and modified becoming Fisher-Kolmogorov Modified Model, where is classified as reaction-diffusion equation which is formed from diffusion equation and logistic equation with involving harvest effect as a reaction term. Both of those terms and each of the parameters on it have important roles that study the solution trajectories, both analytical and numerical. Numerical analysis and its simulation for this model solution are worked using explicit finite difference based on initial conditions."

"Topics in fractional differential equations is devoted to the existence and uniqueness of solutions for various classes of Darboux problems for hyperbolic differential equations or inclusions involving the Caputo fractional derivative. ​​Fractional calculus generalizes the integrals and derivatives to non-integer orders. During the last decade, fractional calculus was found to play a fundamental role in the modeling of a considerable number of phenomena, in particular the modeling of memory-dependent and complex media such as porous media. "

"Homotopy Analysis Method in Nonlinear Differential Equations" presents the latest developments and applications of the analytic approximation method for highly nonlinear problems, namely the homotopy analysis method (HAM). Unlike perturbation methods, the HAM has nothing to do with small/large physical parameters. In addition, it provides great freedom to choose the equation-type of linear sub-problems and the base functions of a solution. Above all, it provides a convenient way to guarantee the convergence of a solution. This book consists of three parts. Part I provides its basic ideas and theoretical development. Part II presents the HAM-based Mathematica package BVPh 1.0 for nonlinear boundary-value problems and its applications. Part III shows the validity of the HAM for nonlinear PDEs, such as the American put option and resonance criterion of nonlinear travelling waves. New solutions to a number of nonlinear problems are presented, illustrating the originality of the HAM. Mathematica codes are freely available online to make it easy for readers to understand and use the HAM. "