Fourier methods in science and engineering This innovative textbook applies the Modified Fourier Series to a variety of problems commonly encountered within science and engineering equipping the reader with a clear pathway through which to use the Fourier method as a solution technique for a wide range of differential equations and boundary value problems. Beginning with an overview of the conventional Fourier series theory the book goes on to introduce the modified Fourier series (MFS) emphasising its notable rate of convergence when compared to traditional Fourier series expansions. It particularly focuses on acoustics elasticity structural vibrations and coupled vibro acoustics. Systematically presenting MFS as a powerful and unified solution method for ordinary differential equations partial differential equations and boundary value problems the book goes on to expand on boundary value problems diving into hot topics such as multi domain multi physics and multi scale characteristics. It will include exemplary MATLAB code to aid students in practically applying MFS. The book will provide students with a comprehensive foundation necessary when solving a wide range of mathematical problems key to engineering applications. It will be of interest to Mechanical engineering and engineering mathematics students