An Analytical Approach to Solving Partial Differential Equations in Mathematical Modeling

Partial Differential Equations (PDEs) are quite central in the applied mathematical modeling, and bring out the essence of yielding solution insights of physics, civil and mechanical engineering, biology and even financial related systems. This study focuses on putting into practice first fundamentals of PDEs such as separation of variables, Fourier, transforms, Green’s functions as well as series expansions. The analytical techniques for solving problems of heat conduction, wave propagation and fluid dynamics are derived and applied and various problems are solved. Comparisons to other methods with numbers bring into light primary and secondary advantages and disadvantages of analytical solutions and their computational complexity. The work focuses on the stability/ convergence/ applications/ potential of these techniques and the author suggests mixed analytical-numerical approaches in further research. This work links abstract theories to real-life problems and ensures better prediction as well as innovations for a variety of fields.