Two areas of mathematics, operational calculus, and integral calculus, have numerous uses in a variety of fields. By discovering solutions to diverse physical and technical systems and simplifying mathematical equations, they play a crucial part in comprehending and resolving real-world problems. The Laplace transform method and the application of Green's functions are two of the main operational calculus techniques. These techniques are especially helpful for resolving boundary value issues, solving linear differential equations with constant coefficients, and simulating physical processes including heat transfer, fluid movement, and electromagnetic fields. On the other hand, the technique of residues and the method of integration by parts are the two main approaches to integral calculus. These techniques are particularly helpful for solving differential equations as well as for evaluating integrals that cannot be assessed by other techniques like substitutions or integration by substitution. The Fourier transform, the Z-transform for operational calculus, the technique of partial fractions, and the method of residues for integral calculus are examples of secondary methods. These techniques may be combined with the fundamental techniques to better comprehend physical and natural events. They are also effective tools for solving specific kinds of integral and differential equations.

Operational, integral calculus, Laplace, Fourier transform.