An Approximate Analytic Solution for the Multidimensional Fractional-Order Time and Space Burger Equation Based on Caputo-Katugampola Derivative

The Burger equation has been widely used to study nonlinear acoustic plane waves in gas-filled tubes, waves in shallow water, and shock waves in gas. Recently, a more comprehensive version of the equation, known as the fractional-order Burger equation, has emerged. However, finding a closed-form for approximate analytic solution using analytical methods for this type of equation is challenging. This paper focuses on researching the multidimensional fractional-order time and space Burger equation based on the Caputo-Katugampola derivative. An approximate analytic solution is obtained using the generalized Laplace homotopy perturbation method. The coefficients of the approximate analytic solution have a recurrence relation similar to the Catalan number in number theory, and the closed form of the approximate analytic solution can be obtained using number theory knowledge. It is worth noting that the Caputo-Katugampola derivative can be reduced to the Caputo derivative, and hence, the closed form for the approximate analytic solution of the multidimensional fractional-order time and space Burger equation based on the Caputo derivative is also obtained.