In this paper, we are presenting our work where the noninteger order partial differential equation is studied analytically and numerically using the noninteger power series technique, proposed to solve a noninteger differential equation. We are familiar with a coupled system of the nonlinear partial differential equation (NLPDE). Noninteger derivatives are considered in the Caputo operator. The fractional-order power series technique for finding the nonlinear fractional-order partial differential equation is found to be relatively simple in implementation with an application of the direct power series method. We obtained the solution of nonlinear dispersive equations which are used in electromagnetic and optics signal transformation. The proposed approach of using the noninteger power series technique appears to have a good chance of lowering the computational cost of solving such problems significantly. How to paradigm an initial representation plays an important role in the subsequent process, and a few examples are provided to clarify the initial solution collection.

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A large class of complexities in mathematical physics, applied mathematics, and engineering are expressed as differential equations with few additions and certain conditions. This research article studies the solution of Volterra integral equations of the second kind where bulge functions take as a known function. To obtain an analytical solution, this study uses the Aboodh transform, the Aboodh inverse transform and the convolution theorem whereas it would be required to discover the precise solution of VIEs. We will also compare it with a numerical solution using a modified Simpson method, and finally, we will represent it graphically.

Nucleate boiling is used in numerous engineering applications, such as the chemical, manufacturing, thermal, nuclear, and electronic industries. This research paper deals with the numerical analysis of bubble growth using a fluid flow model. This physical phenomenon of bubble growth has not been discussed mechanically and does not throw light on empirical models. We are discussing this phenomenon in another way to get the required results in this paper. Simulation of bubble growth is already published by measuring the volume of fluid flow; the method is known as VOFF tracking method. Lee’s model has already discussed the phase change that occurs due to evaporation and condensation of the fluid. We have used the method in which the equation terms involving energy and mass source caused by phase change are incorporated into the control equations by additional subroutines written in C language. We have mentioned in detail the results of simulating mass transfer caused by phase change and the effect of subcooling on bubble growth. The results thus obtained show that the subcooling effect prevents the growth of bubbles from growing due to the certain amount of bubble caps in the subcooled area. The effect of evaporation of the liquid increases the size of the bubbles in both the subcooled and the superheated zone.