Solution of mathematical model for gas solubility using fractional-order Bhatti polynomials

Abstract

Solutions of a mathematical model for gas solubility in a liquid are attained employing an algorithm based on the generalized Galerkin B-poly basis technique. The algorithm determines a solution of a fractional differential equation in terms of continuous finite number of generalized fractional-order Bhatti polynomial (B-poly) in a closed interval. The procedure uses Galerkin method to calculate the unknown expansion coefficients for constructing a solution to the fractional-order differential equation. Caputo?s fractional derivative is employed to evaluate the derivatives of the fractional B-polys and each term in the differential equation is converted into a matrix problem which is then inverted to construct the solution. The accuracy and efficiency of the B-poly algorithm rely on the size of the basis set as well as the degree of the B-polys used. The fractional-order B-Poly technique has been applied to the mathematical model for a gas diffusion in a liquid with gas volume functions f (t)=1−t 1/2 and f (t)=1−t 3/2 . The solutions of the model were obtained which converged with a small number of B-polys basis set. In case of the power series solution, the solution did not converge due to alternating terms present in the solution. We used a Pade approximant on the power series solutions to extract the useful information which showed the solutions are convergent and those solutions were compared with the solutions obtained from the B-poly approach. Excellent agreement was found between the solutions. A Pade approximant was not used on the B-poly solutions because those were convergent with a smaller number of B-polys.

Approximate Solutions of Nonlinear Partial Differential Equations Using B-Polynomial Bases

“…The flawless integration and differentiation are the nature of B-polys that aid in the use of symbolic programming languages, such as Mathematica or Maple. Over any closed interval, B-polys are smooth functions that provide the basis to represent an arbitrary function to desirable correctness [1][2][3]. The specific details and properties of the B-polys are provided in our previous works [1][2][3].…”

“…Over any closed interval, B-polys are smooth functions that provide the basis to represent an arbitrary function to desirable correctness [1][2][3]. The specific details and properties of the B-polys are provided in our previous works [1][2][3]. Briefly, B-polys of nth degree are defined as B i,n (x) = n i…”

“…In the earlier years, various methods were employed to solve linear and nonlinear differential equations, including fractional-order differential equations [4][5][6][7][8][9][10][11][12][13]. In the articles [1][2][3], the authors have solved various differential equations employing a concoction of operational matrix and B-poly basis set. In the earlier progression, the authors used Fractal Fract.…”

“…An excellent agreement has been found between the exact and estimated solutions. In the previous work, following the notation in [1][2][3]13], the work has been protracted to include complete B-poly bases sets that were involved to approximate results of a variety of differential equations [13,16].…”

Technique to Solve Linear Fractional Differential Equations Using B-Polynomials Bases

“…In this study, we are going to implement the modified fractional-order Bhatti polynomial (B-poly) technique [31][32][33][34][35][36][37] that is significantly able to solve a variety of multivariable linear fractional-order differential equations. We chose the fractional-order B-poly due to its well-defined basis set and precision [33].…”