Converting a Nonlinear Differential Equation With Arbitrary Orders to a System

In the research, the Adomian decomposition method will be applied to get the solution of differential equations of arbitrary orders by transforming them into systems. The existence of a unique solution will be proven. The series so-lution converges is discussed, and more than the error is estimated. Some applications will be solved, which are the Basset problem and the Bagley-Torvik equation.


Introduction
D ifferential equations of arbitrary orders have many applications in engineering; some of them are electrical networks (Miller and Ross, 1993;Podlubny, 1999), fluid flow, electromagnetic theory (Kilbas et al., 2006;Kumar et al., 2022), control theory (Daraghmeh et al., 2020a;AbboubakarHamadjam et al., 2021), viscoelasticity, potential theory (Khan et al., 2016;Atangana and Alabaraoye, 2013;Hammad and De la Sen, 2021), neural network systems, and optical systems (Rida and Arafa, 2011;Daraghmeh et al., 2020b).Here, these types of equations will be solved by using the Adomian method (ADM) (Adomian, 1983(Adomian, , 1986(Adomian, , 1995)).There are many advantages to this method in that it solves different types of equations, whether they are linear or nonlinear, in both deterministic and stochastic fields (Adomian, 1989;El-kalla, 2007;Cherruault et al., 1995).In addition, it gives an analytical solution for these equations that does not contain linearization or discretization (Shawaghfeh, 2002;Elkalla, 2008).The convergence of the series will be discussed.The error will be estimated.To clarify that this method is effective, examples and applications (the Bagley-Torvik equation and the Basset problem) will be provided.

Problem formulation
Our problem will be in the following form: x ðpÀ1Þ ð0Þ ¼ k pÀ1 ; p¼ 1; 2; …; q Such as a p and k p be constants, p ¼ 0; 1; 2; …; q À 1.The solution of equations ( 1) and ( 2) can be calculated by reducing the problem to a system of ordinary and fractional equations, each of order g and 0 < g 1.
To convert equation (1) to a system, let Then Which defines this nonlinear system in the following form: Where x ðbÞ of Caputo derivative.The nonlinear system (3) can be written in a compact form as,

;
Taking the inverse operator J g to equation (4) where, If XðtÞ is bounded c t2f ¼ ½0; T, T2R þ , and let jgðtÞj N for all 0 t t T, N is a finite constant, and f ðxÞ satisfy Lipschitz condition such as, In addition, it has an Adomian polynomial representation: Where, Applying ADM to equation ( 5), we get Where, K ¼ ½k 0 ; k 1 ; …;k nÀ1 T and Finally, the solution is:

Existence of a unique solution
Taking the mapping, J : E/E and E is the Banach space (C ðmÞ ðfÞ; k ,k) where C ðmÞ ðfÞ is a class of all continuous vectors X ¼ with norm kXk ¼ P n r¼1 max t2f jx r ðtÞj.
The system (4) has a unique solution if 0 < 4 < 1 and 4 ¼ h Proof: Equation ( 4) can be written as, The mapping J : E/E is defined as If 0 < 4 < 1 then the mapping J is contraction, then there exists a unique solution to the problem (1-2).

Convergence of the series solution
Theorem 2. The series solution (11) converges if jX 1 j < ∞, 0 < 4 < 1; Proof: Define a sequence fS r g such that S r ¼ P r n¼0 X n ðtÞ is the sequence of partial sums from the series solution P ∞ n¼0 X n ðtÞ; then (El-kalla, 2007), Let S r and S w be two partial sums with r > w.We aim to prove that fS r g is a Cauchy sequence in this Banach space.
ET g GðgÞ Using the triangle inequality, we get If jX 1 ðtÞj < ∞ and v/∞ therefor, kS r À S w k/0 and fS r g is a Cauchy sequence in this Banach space; hence the series P ∞ n¼0 X n ðtÞ converges.

Error estimation
Theorem 3: The maximum absolute error of the series solution ( 11) can be estimated as Proof: Using Theorem 2, we get: But, S r ¼ P r n¼0 X n ðtÞ as r/∞ then, S r /XðtÞ so, Then, the maximum absolute error can be estimated to be
From this example, we see that ADM gives the exact solution and makes the computations easy when the problem is converted to a system.This clarifies the effectiveness of this algorithm when solving nonlinear multi-term FDEs.

Conclusions
In this research, we use ADM to solve multi-term FDE by transforming it into a system.We see from the given examples that the transformation made the equations simpler and easier in calculations.In addition, sometimes it can give the exact solution directly, as illustrated in Examples 1.