The new Laplace variational iterative method is used in this research for solving the (2+1)-D and (3+1)-D Burgers equations. This technique relies on the modified variational iteration method and the Laplace transform. To apply this approach, the differential problem is first transformed into an algebraic form using the Laplace transform, and then the algebraic equations are iteratively solved using the modified variational iterative approach. By utilizing this technique, the Burgers equations can be solved both numerically and analytically. The study demonstrates the effectiveness of the new Laplace variational iterative approach through three specific examples.

In this research, we presented two classical numerical techniques to solve some oscillatory problems arised in various applications of sciences and engineering. Adomian decomposition method (ADM) and Haar wavelet method (HWM) are utilized for this purpose. Some numerical examples have been performed to illustrate the accuracy of the present methods.

As two-dimensional coupled system of nonlinear partial differential equations does not give enough smooth solutions, when approximated by linear, quadratic and cubic polynomials and gives poor convergence or no convergence. In such cases, approximation by zero degree polynomials like Haar wavelets (continuous functions with finite jumps) are most suitable and ...

Many real-world phenomena such as nerve pulse transmission, fluid transport, and chemical kinetics are modeled using nonlinear partial differential equations with time-fractional derivatives. Among them, the time-fractional Burgers-Huxley equation plays a significant role due to its ability to capture both diffusion and reaction mechanisms with memory effects. Solving such equations in higher dimensions is highly challenging and calls for efficient analytical approaches. In this work we present a technique for handling the time-fractional Burgers-Huxley equation up to k-dimensions by employing Laplace transform based homotopy perturbation method (LT-HPM). The LT-HPM is adopted based on its advantage in dealing with nonlinear terms and simplify the solution process by converting fractional derivatives into more manageable expressions. Unlike other hybrid approaches, LT-HPM is less computational complex and provides rapid convergent series solutions without requiring any linearization or restrictive assumptions. To showcase the effectiveness of this approach, we solve a pair of examples: a 2-D case and a 3-D case. The obtained results confirm that LT-HPM is accurate and powerful in tackling complex nonlinear PDEs in higher dimensions.

In this research paper, an investigation of charge and current flow in an LCR series circuit has been presented. For this purpose, basis functions of Chebyshev wavelets of the second kind have been utilized. The proposed method involves representing the highest-order derivatives as a series of basis functions using Chebyshev wavelets. In order to demonstrate the effectiveness of this approach, numerical examples have been provided, and their results are presented to show the accuracy of the proposed scheme.