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While compiling this book, I have developed chapter one on 'partial differential equations', chapter two on 'The coordinates systems and orthogonality', chapter three on 'Orthogonal basis and Hilbert space', chapter four on 'Adomian Decomposition method and Laguerre polynomials, chapter five on 'Fourier series and Fourier-Bessel series expansions (Harmonic Analysis)', chapter six on 'Series Solution and Orthogonal Polynomials', chapter seven on 'the Integral' and 'Laplace Transforms', chapter eight on 'Exact solutions for the wave equations', chapter nine on 'Exact solutions for the diffusion equations and chapter ten on 'Exact solutions for the Laplace's equations'. Partial Differential Equations are the basis of all physical problems. Real world problems can be formulated as partial differential equations (initial-boundary value problems). For most of these differential equations, exact solutions are not known. Based on the success of Fourier analysis and Hilbert space theory, orthogonal basis functions expansions undoubtedly count as fundamental concept of mathematical analysis.