Residual power series method

The residual power series method (RPSM) is a semi-analytical technique fitted to solve ordinary and partial differential equations of different types, classifications, and orders side by side with some related theories, such as, existence, uniqueness, and stability. The RPSM employs a convergent series solution by minimizing a residual error in a truncated series solution. This method can be used to construct power series expansion solutions for strongly linear and nonlinear equations without linearization, perturbation, discretization, or restrictive assumptions. Different from the classical power series method, the RPSM does not need to compare the coefficients of the corresponding terms and a recursion relation is not required. In this approach the method computes the coefficients of the power series by a chain of algebraic equation(s) of one or more variable(s) depending on the solution(s) domain.

Multiple fractional power series

The RPSM is used to seek a multiple fractional power series (MFPS) solution to fractional differential equations. This procedure can be achieved by substituting its MFPS expansion among its truncated residual error function. From the resulting equation a recursion formula for the computation of the coefficients is derived, while the coefficients in the MFPS expansion can be computed recursively by recurrent fractional differentiation of the truncated residual error function by means of the symbolic computation software used.

The refinement forms of the RPSM present the computational solution of the fractional differential equations subject to given constraints conditions based on the generalized Taylor series formula. The algorithm methodology is based on constructing a MFPS solution in the form of a rapidly convergent series with minimum size of calculations. On the other aspect as well, MFPS solution guarantee the procedure algorithm of the RPSM and illustrate its theoretical statements in term of potentiality, generality, and superiority for solving various fractional differential and integral equations. In fact, the main advantage of this new algorithm is the simplicity in computing the coefficients of terms of the series solution by using the differential operators only and not as the other well-known analytic techniques that need the integration operators which is difficult in the fractional case. Moreover, the refinement algorithm can be easily applied in the spaces of higher dimension solution and can be applied without any limitation on the nature of the equation and the type of classification.

Importance of the power series solution

Series expansions are very important aids in numerical calculations, especially for quick estimates made in hand calculation, for example, in evaluating functions, integrals, or derivatives. Solutions to ordinary and partial differential equations of different types, classifications, and orders, can often be expressed in terms of series expansions. Since, the advent of computers, it has, however, become more common to treat such equations directly, using different approximation method instead of series expansions. But in connection with the development of automatic methods for formula manipulation, one can anticipate renewed interest for series methods. These methods have some advantages, especially in multidimensional solutions for different categories of equations of classical, fuzzy, and fractional order derivatives and integrals.

Characteristics

The RPSM was first devised in 2013 by the Jordanian mathematician Omar Abu Arqub of Al-Balqa` Applied University as an efficient method for determining the values of the coefficients of the power series solution of fuzzy differential equations. In 2014 the RPSM was modified to solve fractional forms of differential equations by connecting it with new general form of MFPS.

The RPSM distinguishes itself from various other analytical and numerical methods in several important aspects. Firstly, it is a series expansion method that is not directly dependent on small or large physical parameters. Thus, it is applicable for not only weakly but also strongly nonlinear problems, going beyond some of the inherent limitations of the standard perturbation methods. Secondly, the RPSM gives excellent flexibility in the expression of the solution and how the solution is explicitly obtained. Thirdly, unlike the other analytic approximation techniques, the RPSM provides a simple way to ensure the convergence of the series solution by minimizing the related residual error. Fourthly, in the RPSM, it is possible to pick any point in the interval of integration and keep the approximate solutions and their derivatives applicable. Fifthly, the RPSM does not require discretization of the variables, and it is not affected by computational rounding errors and does not require large computer memory and time. Sixthly, it obtains global solutions applicable to mathematical, physical, and engineering problems. Seventhly, the RPS method does not require any converting while switching from the low-order to the higher-order and from simple linearity to complex nonlinearity; as a result the method can be applied directly to the given problem by choosing an appropriate initial guess approximation.

The RPSM is also able to combine with other techniques employed in nonlinear differential equations, such as spectral methods and Padé approximant methods. It may further be combined with computational methods, such as the boundary element method and reproducing kernel Hilbert space method, to allow the linear method to solve nonlinear systems.

Applications

In the last few years, the RPSM has been applied to solve a growing number of nonlinear ordinary and partial differential equations of different types, classifications, and orders, in science, finance, and engineering. It has been successfully applied in the numerical solution of the generalized Lane–Emden equation, which is a highly nonlinear singular differential equation, in the numerical solution of higher-order regular differential equations, in predicting and representing the multiplicity of solutions to boundary value problems of fractional order, in the solution of composite and non-composite fractional differential equations, in the mathematical derivation of a new general form for Taylor's formula side by side with some physical applications, in constructing and predicting the solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations, in approximate solution of the nonlinear fractional KdV–Burgers equation, in the numerical solutions for the wave, telegraph, Poisson, and Navier–Stokes equations of time-fractional or space-fractional derivative, in the series solution of the fractional Fisher's equation, in the series solution of the fractional foam drainage equation, in the numerical solution of systems of differential equations, in the numerical solution of systems of singular differential equations, in the representation of the exact solutions for algebraic differential algebraic systems, in the solution of the fractional Sharma–Tasso–Olever equation, in the solution of time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves, in the solution of fractional Brownian motion models, in construction of fractional power series solution to fractional Boussinesq equations, in the solution of nonlinear time-dependent generalized FitzHugh–Nagumo equations, and in the series solution of time-space-fractional Benney–Lin equations arising in falling film problems.

Software libraries

Numerical techniques are widely used by scientists and engineers to solve their problems. A major advantage for numerical techniques is that a numerical answer can be obtained even when a problem has no analytical solution. However, result from numerical analysis is an approximation, in general, which can be made as accurate as desired. Because a computer has a finite word length only a fixed number of digits are stored and used during computation.

The RPSM is an analytic approximation method designed for the computer era with the goal of computing with functions instead of numbers. In conjunction with a computer algebra system such as Mathematica, Mathcad, Maple, or Matlap, one can gain analytic approximations to arbitrarily nonlinear problems by means of the RPSM in only a few seconds. In fact, a simple software package code can be written for solving such problems with singularities, multiple solutions, and multipoint boundary conditions in either a finite or an infinite interval, and includes support for certain types of nonlinear equations.

Brief description

For convenience, the reader is asked to refer to the Applications Section in order to know more details about the RPSM, including their construction, their motivation for use, their characteristics over conventional method, and their applications for solving different categories of linear and nonlinear differential and integral equations of different types, classifications, and orders.

Here, the main steps in the algorithm of the RPSM: Firstly, define the residual error function. Secondly, replace the dependent variable and all its derivative by the MFPS expansion. Thirdly, apply the recurrent differentiations for Step 2. Fourthly, solve the algebraic equations obtained from Step 3 to generate the unknown coefficients for the MFPS expansion. Fifthly, substitute the coefficients generated from step 4 in the MFPS expansion to find the require RPSM solution.

This procedure can be repeated till the arbitrary order coefficient of RPSM solution is obtained. Moreover, higher accuracy can be achieved by evaluating more components.

Future recommendations

Numerical methods for the solutions of ordinary and partial differential equations of different types, classifications, and orders are essential for the analysis of physical, mathematical, and engineering phenomena. Strong solvers are necessary when exploring characteristics of such equations. Here, the RPSM is introduced as strong novel solver for some certain types of such problems to enlarge its applications range. It is analyzed that the proposed algorithm is well suited for use in differential and integral operator's problems of volatile orders and resides in its simplicity in dealing with various constraints conditions. It is worth to be pointed out that the RPSM is still suitable and can be employed for solving different strongly linear and nonlinear problems.

References

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