[Paper Review] An application of Bell polynomials in numerical solving of nonlinear differential equations

Josef Rebenda|arXiv (Cornell University)|Jan 27, 2019

Iterative Methods for Nonlinear Equations4 citations

TL;DR

This paper proposes a novel application of partial ordinary Bell polynomials to simplify the numerical solution of nonlinear differential equations via the differential transformation method. By reformulating Faà di Bruno’s formula using Bell polynomials, the method enables efficient computation of nonlinear terms without symbolic derivatives, yielding accurate series solutions through recursive arithmetic operations, as demonstrated in two initial value problems with exact solutions identified via pattern recognition.

ABSTRACT

Partial ordinary Bell polynomials are used to formulate and prove a version of the Fa\\`{a} di Bruno's formula which is convenient for handling nonlinear terms in the differential transformation. Applicability of the result is shown in two examples of solving the initial value problem for differential equations which are nonlinear with respect to the dependent variable.

Motivation & Objective

• To address the computational complexity of handling nonlinear terms in differential transformation methods, especially those involving composite functions of the unknown solution.
• To develop a systematic and computationally efficient approach for transforming nonlinear terms in differential equations using combinatorial properties of Bell polynomials.
• To provide a practical alternative to existing semi-analytical methods like Adomian decomposition, which require symbolic derivatives or complex integral computations.
• To demonstrate that the proposed method yields accurate series solutions through recursive arithmetic operations, with potential for closed-form identification.

Proposed method

• The paper introduces a modified version of Faà di Bruno’s formula using partial ordinary Bell polynomials to express higher-order derivatives of composite functions.
• It formulates the differential transformation of a nonlinear function $ f(u(t)) $ as a sum involving Bell polynomials of the transformed coefficients $ U(k) $, eliminating the need for symbolic derivatives of $ f $.
• The method relies on recursive computation of $ U(k) $ using previously computed $ U(i) $ for $ i < k $, ensuring computational efficiency.
• The algorithm uses the inverse differential transformation to reconstruct the solution as a Taylor-like series, with coefficients derived from Bell polynomial expansions.
• The approach avoids numerical integration, differentiation, or iterative solvers, relying solely on basic arithmetic operations.
• The method is validated on two initial value problems, showing convergence to exact solutions through identifiable coefficient patterns.

Experimental results

Research questions

• RQ1 Can partial Bell polynomials be used to streamline the differential transformation of nonlinear terms in ODEs without symbolic differentiation?
• RQ2 How can Faà di Bruno’s formula be reformulated using Bell polynomials to improve computational efficiency in semi-analytical methods?
• RQ3 To what extent can this method produce exact solutions in closed form when applied to nonlinear initial value problems?
• RQ4 What is the computational advantage of this approach over existing methods like Adomian decomposition or homotopy perturbation in terms of required operations?

Key findings

• The differential transformation of a nonlinear term $ f(u(t)) $ is successfully expressed using partial ordinary Bell polynomials, enabling direct computation from the transformed coefficients $ U(k) $ without symbolic derivatives.
• For the first example, the solution coefficients follow the pattern $ U(k)[0] = 0 $ for even $ k $, and $ U(k)[0] = (-1)^{(k-1)/2} \frac{2^k}{k!} $ for odd $ k $, leading to the exact solution $ u(t) = \sin 2t $.
• In the second example, the method produced coefficients matching the Taylor series of $ \sin 2t $, confirming the exact solution through pattern recognition.
• The algorithm is computationally efficient, relying only on arithmetic operations and recursive computation, avoiding integrals, derivatives, or initial guess requirements.
• The method enables identification of exact solutions in closed form when the coefficient pattern matches a known function, enhancing accuracy beyond standard numerical approximations.
• The approach is generalizable to other problems, including boundary value problems, and offers a function-analytic approximation that preserves analyticity near the expansion point.