Solution to Laplace’s Equation Using Quantum Calculus

Abstract

The quantum calculus emerged as a new type of unconventional calculus relevant to both mathematics and physics. The study of quantum calculus or q-calculus has three hundred years of history of development since the era of Euler and Bernoulli, and was appeared as one of the most arduous techniques to use it in mathematics as well as physical science. At present, it is used in diverged mathematical areas like number theory, orthogonal polynomials, basic hypergeometric functions, etc. Furthermore, in order to get analytical approximate solutions to the ordinary as well as partial differential equations, q-reduced differential technique and quantum separation of variable technique are used in mathematics, Mechanics, and physics. In this paper, Laplace’s equation, a well-known equation in both Physical and Mathematical sciences, has been solved extensively based on the basics of calculus, transformation methods, and q-separation of variable method. In addition, solutions to the Laplace’s equation as obtained by using different boundary conditions are revisited and reviewed. Consequently, all the necessary basics of q-calculus are displayed one by one, and thereafter, the process of finding its solution in view of quantum calculus is described extensively. In order to find out the exact solutions the dimensionality of all the parameters related to the problem has been described. As an essential outcome, it is also found that, as q tends to 1, the solution takes the form as it is in general physics. Hence, this article presents a review and extension that describe the solution to Laplace’s equation in view of both Leibnitz and quantum calculus. Thus, it can add a pedagogical exercise for the students of both physical and mathematical sciences to understand the usefulness of quantum calculus.