Fractional order differential equations are generalizations of ordinary differential equations where the order of the derivative can be any real number, not just an integer. The concept of fractional derivatives might seem counterintuitive, but its origins can be traced back to the 17th century when Leibniz and L&rsquoHospital first pondered the possibility of derivatives of non-integer order. Mathematicians like Euler, Laplace, Fourier, and Riemann later contributed to the development of fractional calculus. However, it was only in recent decades that the practical applications of fractional order differential equations gained significant attention. Unlike ordinary derivatives, which involve integer-order differentiation, fractional derivatives capture the behavior of a function over its entire history. This property is crucial for modeling systems with memory effects. There are several definitions of fractional derivatives available in the literature, but a few are widely recognized, such as the modified Riemann-Liouville, Caputo, Grünwald-Letnikov, and local fractional derivatives. In this thesis, we have used the Caputo and local fractional derivatives. A key concept in the development of fractional calculus is the Riemann-Liouville definition of fractional differentiation. However, the precise applicability of the Riemann-Liouville derivative is limited since the derivative of the constant is not zero. In certain contexts, the standard Riemann-Liouville derivative may not be as suitable for certain scientific and engineering applications as the Caputo fractional derivative. The Riemann-Liouville derivative is often harder to interpret physically, whereas the Caputo derivative incorporates initial conditions of integer order, which are more practical for real-world scenarios. This thesis examines fractional order problems in various fields, such as wave equations, heat equations under both precise and uncertain conditions, telegraph equations, financial systems, and biological systems.