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REVIEW 5 major objections 3 minor 43 references

Adomian decomposition method reformulated using dimensionless nonlinear perturbation theory

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Pith's one-line read The paper claims that a dimensionless perturbation reformulation of the Adomian decomposition method, centered on the recurrence (25), resolves the long-standing mismatch in the order of the expansion parameter.

desk verdict The paper's proposed resolution of Adomian's order-mismatch dilemma fails because the commutativity identities (26)–(27) behind the new recurrence (25) are false for nonlinear operators, and the remaining normalization argument is circular. read the letter →

arxiv 2501.10398 v1 pith:AAEWZX7G submitted 2025-01-01 physics.comp-ph

Albert S. Kim This is my paper · ORCID
classification physics.comp-ph
keywords AdomiandecompositionmethodnonlinearperturbationtheoryexpansionparameterorderdimensionlessanalysisrecurrenceformulaLyapunovstabilitycommutativeoperatorsdifferentialequations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to resolve the long-standing inconsistency in the Adomian decomposition method (ADM) between the order of the expansion parameter used to build Adomian polynomials and the order of the recurrence that generates the series terms. It reformulates the method as a dimensionless nonlinear perturbation theory, setting the dimensionless time scale equal to the expansion parameter $\lambda$, and derives a new recurrence, Eq. (25), in which $\lambda$ appears explicitly and uniformly. If the derivation holds, the expansion parameter acquires a physical meaning and the mismatch identified in earlier work disappears, opening a cleaner route to higher-order ADM terms. The argument stands or falls on a commutation claim about the inverse operator and the nonlinear polynomial, which is stated but not proved.

What carries the argument

The load-bearing object is the dimensionless perturbation series $u(\lambda)=\sum_{n=0}^\infty \lambda^n u_n$, combined with the normalizations $\bar U=1$ and $t_0^s=\lambda$, which converts the governing equation into a standard perturbation hierarchy. The recurrence (25) is produced by applying the inverse operator $L^{-1}$ termwise and assuming the commutation identities $L^{-1}R[u_k]=R[L^{-1}u_k]$ and $L^{-1}A_k[\{u_{j\le k}\}]=A_k[L^{-1}\{u_{j\le k}\}]$ (Eqs. (26)-(27)); the second identity is what lets the nonlinear Adomian polynomial be evaluated after integration. This step is essential because without it the new recurrence does not follow from the perturbation hierarchy.

What would settle it

Take $f(u)=u^2$ with $u_0=t$ and $u_1=t$. The first Adomian polynomial is $A_1=2u_0u_1=2t^2$, and the claimed identity (27) would require $L^{-1}(2t^2)=2(\int_0^t u_0\,dt)(\int_0^t u_1\,dt)$, i.e. $\frac{2}{3}t^3=\frac{1}{2}t^4$, which is false. Checking this one equality for any simple nonlinearity settles the central claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that Adomian's dilemma comes from using $\lambda$ only as a temporary bookkeeping device when constructing the Adomian polynomials, while deriving the recurrence without any $\lambda$. After nondimensionalizing the governing equation and choosing the dimensionless time scale as $t_0^s = \lambda$, the paper obtains the recurrence $$u_{k+1} = -\$\lambda$ R[$L^{{-1}}$(u_k)] - \$\lambda$ A_k($L^{{-1}}$\{u_{j\le k}\}), \tag{25}$$ in which every term is visibly of matched order in $\lambda$. The paper claims this recurrence, together with the commutation relations (26)-(27), resolves the order mismatch and ties ADM to standard perturbation theory with a physically dimensioned parameter.

Load-bearing premise

The derivation requires that the inverse integral operator $L^{-1}$ can be pushed inside the nonlinear Adomian polynomial, so that integrating the polynomial equals evaluating the polynomial on integrated components; this equality does not hold for ordinary polynomial nonlinearities such as $u^2$.

Editorial extensions

If this is right

  • If Eq. (25) is correct, every component $u_{k+1}$ carries exactly one more power of $\lambda$ than $u_k$, so the final series is order-consistent in the expansion parameter.
  • The expansion parameter acquires a physical dimension as $t_0^s$, connecting ADM's abstract $\lambda$ to a real time scale in transport and wave problems.
  • The scaling property (13) extends the construction to nonlinearities involving $U$, $\dot U$, and $U'$, so derivative-coupled nonlinear terms are covered by the same dimensionless ordering.
  • Truncating the series and setting $\lambda=1$ gives an analytical approximation whose terms are synchronized, which the paper argues improves the basis for convergence and accuracy of ADM solutions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The commutation identity (27) is not generally valid for polynomial nonlinearities, so the new recurrence (25) is not equivalent to the standard ADM recurrence unless the class of nonlinearities is restricted; the paper does not state such a restriction.
  • If $L^{-1}$ is kept outside the Adomian polynomial, the standard recurrence (6) follows instead, suggesting the 'dilemma' may be a bookkeeping artifact of where $L^{-1}$ is placed rather than a physical inconsistency.
  • A direct numerical test would compare the series generated by (25) with the standard ADM series for a nonlinear oscillator or a convection-diffusion-reaction equation with a known exact solution; the two series already differ at the $u_1$ term whenever the nonlinearity is not linear.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 3 minor

Summary. The paper proposes a reformulation of the Adomian decomposition method (ADM) using dimensionless nonlinear perturbation theory. The author identifies an 'Adomian dilemma': the recurrence derived from the decomposition (Eq. 6) is not reproduced if a perturbation-style series with an expansion parameter λ is substituted directly into the governing equation. The paper attempts to resolve this by nondimensionalizing the equation, introducing representative scales Ū and t0, assuming a scaling law for the Adomian polynomials (Eq. 13), and deriving a new recurrence (Eq. 25) via commutativity of the inverse operator with the remainder operator and the Adomian polynomials (Eqs. 26-27). The paper also claims connections to Lyapunov's stability theory and concludes that the order mismatch is resolved. No worked example or numerical validation is provided.

Significance. If the derivation were correct, the paper would offer a useful clarification of the relationship between ADM and perturbation theory. The literature review is reasonable, and the identification of the order-mismatch issue is a genuine topic in the ADM literature. However, the central derivation rests on several unproven and, in general, false identities, and the normalization steps eliminate the physical content they claim to introduce. The paper makes falsifiable claims but supplies no numerical or analytical validation. The strength of the paper is its clear statement of the problem; the weakness is that the proposed resolution is not supported by the mathematics presented.

major comments (5)
  1. [§3.2, Eqs. (24)-(25)] The step from Eq. (24) to Eq. (25) is not justified. Eq. (24) states ∂τ^s u_k = -λ R[u_k] - λ A_k({u_j≤k}), so applying L^{-1} to both sides yields an expression for u_k, not u_{k+1}. To obtain u_{k+1}, one would need an equation involving ∂τ^s u_{k+1} on the left, but Eq. (24) has u_k on the right and the same index on the left. The index shift from k to k+1 is asserted, not derived, and this is load-bearing for the claimed resolution.
  2. [§3.2, Eq. (27)] The commutativity identity L^{-1}A_k[{u_j≤k}] = A_k[L^{-1}{u_j≤k}] is false for general nonlinear operators. For example, take N(u)=u^2, so A_1=2u_0u_1. With L^{-1}f=∫_0^τ f, the left side is 2∫_0^τ u_0u_1, while the right side is 2(∫_0^τ u_0)(∫_0^τ u_1); these are generically unequal. Since A_k for k≥1 generally contains products of components, the identity fails for essentially all nonlinearities. Eq. (26) is also false when R contains lower-order derivatives, e.g., R=∂τ gives L^{-1}∂τ u = u-u(0), whereas ∂τ L^{-1}u = u. These identities are central to Eq. (25), so the recurrence is unsupported.
  3. [§3.1, Eq. (13)] The scaling law in Eq. (13) is not generally valid. The notation in Eq. (12) uses p, q, r as component indices in the arguments of A_n, but Eq. (13) turns these same symbols into exponents of the representative scales Ū, V̇, and U′. There is no reason that a sum of terms of different degrees, such as A_2 = f'(u0)u2 + (1/2)f''(u0)u1^2, should scale as a single monomial with p+q+r=n. For a general analytic nonlinearity, A_n contains terms of differing homogeneity, so a unique α_n^* does not exist. This undermines the subsequent nondimensionalization and the claim of a parameter-free resolution.
  4. [§3.2, Eqs. (20)-(22)] The normalizations Ū=1 and t0^s=λ are dimensionally inconsistent and eliminate the physical content the paper claims to introduce. t0^s has dimensions T^s while λ is declared dimensionless, so Eq. (21) cannot hold as an equality of physical quantities. Setting Ū=1 is just a choice of units, and the subsequent λ=1 in Eq. (19) fixes t0=1, removing the representative time scale. The procedure recovers, by construction, the standard perturbation assumption that the nonlinear and remainder terms are multiplied by a small parameter λ; it does not derive this assumption from physical scaling.
  5. [Whole paper] No numerical example or convergence check is provided. Since the central claim is that Eq. (25) resolves the order mismatch and leads to faster convergence and higher accuracy, the absence of even a single worked nonlinear problem (compared with an exact or numerical solution) leaves the practical claims untested. This is particularly important given that the derivation of Eq. (25) is invalid.
minor comments (3)
  1. [Abstract and affiliation] The abstract contains grammatical errors, e.g., 'unresolved issue regarding the mismatch' should be 'an unresolved issue regarding the mismatch', and the affiliation contains a typo: 'Unive rsity'.
  2. [Eq. (18)] The factor t0^s/U^{k-1} preceding A_k in Eq. (18) is not clearly defined for k=0, and its derivation from the nondimensionalization of A_k is not shown; please clarify the notation.
  3. [Section 2 and Appendix] The paper uses 'Lyapunov's stability theory' and 'Lyapunov's artificial small parameter method' as if they were the same thing, but the connection between Lyapunov exponents (Refs. [42,43]) and the formal perturbation parameter is not established; this conflation should be addressed explicitly.

Circularity Check

2 steps flagged · score 7.0 of 10

The claimed resolution of Adomian's dilemma is achieved by defining the expansion parameter through a scale choice and by assuming an unproved commutativity of L^{-1} with nonlinear Adomian polynomials, so the central recurrence is an ansatz rather than a derivation.

  1. self definitional [Section 3.2, Eqs. (20)-(22) and Concluding Remarks]
    "Without loosing generality, we let U = 1 (20) t0^s = λ (21) to rewrite Eq. (18), such as ∞∑_{k=0} λ^k [∂^s u_k/∂τ^s + λR[u_k] + λA_k({u_k})] = gλ (22) which is similar to a standard perturbation series having λ as an expansion parameter."

    The 'resolution' of the order mismatch consists of inserting λ by choosing the dimensionless scales Ū=1 and t0^s=λ. This makes λ appear in front of R and A by construction, not by physical derivation. The parameter is then removed by letting λ=1 ('the truncated form of u is finally obtained by letting λ=1 in Eq. (19)'), so the expansion parameter is defined into the problem and discarded at the end. The claimed reconciliation with perturbation theory is therefore equivalent to the standard perturbation assumption it purports to derive.

  2. other [Section 3.2, Eqs. (24)-(27)]
    "By applying the inverse operator L−1 from the left on the both side of Eq. (24), we obtain a new recurrence formula of Eq. (6), such as uk+1 = −λR[L−1(uk)] − λAk(L−1{uj≤k}) (25) using the commutative relationships of L−1R[uk] = R[L−1uk] (26), L−1Ak[{uj≤k}] = Ak[L−1{uj≤k}] (27)."

    Equation (25) follows from Eq. (24) only if L^{-1} commutes with the Adomian polynomials. This commutativity is not a consequence of the definitions: A_k is a homogeneous polynomial of degree k+1 in u_0,...,u_k, and the s-fold integral L^{-1} does not in general pass through products (e.g., A_1=2u_0u_1 gives L^{-1}A_1 ≠ A_1(L^{-1}u_0, L^{-1}u_1)). Asserting (27) is therefore equivalent to assuming the componentwise-integrated solution form that the recurrence is supposed to produce, so the central recurrence is an ansatz, not a derived resolution. The index shift from u_k to u_{k+1} is also not obtained from Eq. (24).

full rationale

The paper's central claim is that Eq. (25) 'resolves the Adomian's dilemma of the order mismatch of the expansion parameter.' The chain leading to Eq. (25) has two load-bearing moves, both of which import the conclusion rather than derive it. First, Eqs. (20)-(21) set Ū=1 and t0^s=λ, so λ appears in Eq. (22) as a perturbation parameter by definition; the final λ=1 then erases it, leaving the original ADM recurrence structure. Second, Eq. (25) is obtained from Eq. (24) by asserting commutativity (26)-(27); identity (27) is false for nonlinear Adomian polynomials, and no proof or numerical check is supplied. Thus the 'resolution' is an ansatz presented as a derivation. The paper does not fit parameters to data, and the only self-citation (ref. [44]) is an example equation, not load-bearing. The circularity is partial but central: the main result reduces to the assumptions it claims to resolve.

Assumptions & free parameters 3 free parameters · 3 assumptions · 1 invented entities

The central claim rests on two unproved and false commutativity assumptions (Eqs. 26-27), a non-general scaling law (Eq. 13), and ad hoc choices of the representative scales (Ū = 1, t0^s = λ). These are introduced specifically to make the derivation appear to resolve the dilemma, but they do not represent independent physical input.

free parameters (3)
  • Ū (representative magnitude of U) = 1
    Chosen by hand in Eq. (20) to simplify the dimensionless equation; this removes the physical scale rather than deriving it.
  • t0 (representative time scale) = λ^{1/s} (set via t0^s = λ)
    Chosen such that t0^s equals the expansion parameter λ (Eq. 21). Since λ is eventually set to 1, this collapses to t0 = 1, making the nondimensionalization trivial.
  • λ (expansion parameter) = 1 at the end
    Introduced as the perturbation parameter and then set to 1 (Section 3.2), so the 'dimensioned' parameter does no physical work.
assumptions (3)
  • ad hoc to paper L^{-1} R = R L^{-1} for the operator split in Eq. (26)
    Asserted as a commutative relationship, but false when R contains lower-order time derivatives; e.g., for L = d/dt and R = d/dt, L^{-1}R f = f - f(0) while R L^{-1} f = f.
  • ad hoc to paper L^{-1} A_k[{u}] = A_k[L^{-1}{u}] in Eq. (27)
    Assumes the integral operator commutes with nonlinear polynomial evaluation, which is false (integral of a product is not the product of integrals).
  • ad hoc to paper An scales as α_n^* = Ū^p V̇^q Ū'^r with p + q + r = n in Eq. (13)
    Holds only for homogeneous nonlinearities of a specific form; general analytic nonlinearities give An as a sum of terms of varying degree, so the scaling exponent is not n.
invented entities (1)
  • Dimensioned expansion parameter λ defined via t0^s = λ
    purpose: To give the perturbation parameter a physical dimension and justify the ordering that resolves Adomian's dilemma
    No falsifiable prediction or external handle; the parameter is removed by setting λ = 1 at the end, so it does not connect to any measurable quantity.

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Pith. "Pith review of Adomian decomposition method reformulated using dimensionless nonlinear perturbation theory." pith.science (2026). https://pith.science/paper/AAEWZX7G

@misc{pith2026250110398,
  author       = {Pith},
  title        = {Pith review of: Adomian decomposition method reformulated using dimensionless nonlinear perturbation theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AAEWZX7G}},
  note         = {Machine review of arXiv:2501.10398}
}
read the original abstract

The Adomian decomposition method (ADM) is a universal approach to solving governing equations in various engineering and technological applications. The applicability of the ADM is almost limitless due to its universal applicability, but its convergence rate and numerical accuracy are sensitive to the number of truncated terms in series solutions. More importantly, Adomian formalism still holds unresolved issues regarding the mismatch of the order of the expansion parameter. The current work provides an in-depth analysis of Adomian's decomposition method, Lyapunov's stability theory, and the nonlinear perturbation theory to resolve the fundamental mismatch with physical interpretation.

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Works this paper leans on

43 extracted references · 35 canonical work pages

  1. [1]

    Reviews of Modern Ph ysics 35(1), 185–207 (1963) https://doi.org/10.1103/RevModPhys.35.185

    Adomian, G.: Linear Stochastic Operators. Reviews of Modern Ph ysics 35(1), 185–207 (1963) https://doi.org/10.1103/RevModPhys.35.185

    work page doi:10.1103/revmodphys.35.185 1963

  2. [2]

    Journa l of Mathematical Analysis and Applications 55(2), 441–452 (1976) https://doi.org/10.1016/0022-247X(76)90174-8

    Adomian, G.: Nonlinear stochastic differential equations. Journa l of Mathematical Analysis and Applications 55(2), 441–452 (1976) https://doi.org/10.1016/0022-247X(76)90174-8

    work page doi:10.1016/0022-247x(76)90174-8 1976

  3. [4]

    Journal of M athematical Analysis and Applications 76(2), 309–327 (1980) https://doi.org/10.1016/0022-247X(80)90035-9

    Adomian, G., Malakian, K.: Self-correcting approximate solution by the iterative method for linear and nonlinear stochastic differential equations. Journal of M athematical Analysis and Applications 76(2), 309–327 (1980) https://doi.org/10.1016/0022-247X(80)90035-9

    work page doi:10.1016/0022-247x(80)90035-9 1980

  4. [5]

    Mathematics in Science and Eng ineering, vol

    Adomian, G.: Stochastic Systems. Mathematics in Science and Eng ineering, vol. 169. Academic press, New York, London, Paris (1983)

    work page 1983

  5. [6]

    Jou rnal of Mathematical Analysis and Applications 96(2), 330–340 (1983) https://doi.org/10.1016/0022-247X(83)90044- 6 6

    Adomian, G., Bellomo, N., Riganti, R.: Semilinear stochastic systems: Analysis with the method of the stochastic Green’s function and application in mechanics. Jou rnal of Mathematical Analysis and Applications 96(2), 330–340 (1983) https://doi.org/10.1016/0022-247X(83)90044- 6 6

    work page doi:10.1016/0022-247x(83)90044- 1983

  6. [7]

    Journal of Mathematical Analysis and Applications 113(1), 202–209 (1986) https:// doi.org/10.1016/0022-247X(86)90344-6

    Adomian, G.: A new approach to the heat equation - An application o f the decomposition method. Journal of Mathematical Analysis and Applications 113(1), 202–209 (1986) https:// doi.org/10.1016/0022-247X(86)90344-6

    work page doi:10.1016/0022-247x(86)90344-6 1986

  7. [8]

    Mathematical and Computer Modelling 13(7), 17–43 (1990) https://doi.org/10.1016/ 0895-7177(90)90125-7

    Adomian, G.: A review of the decomposition method and some recen t results for nonlinear equations. Mathematical and Computer Modelling 13(7), 17–43 (1990) https://doi.org/10.1016/ 0895-7177(90)90125-7

    work page 1990

  8. [9]

    Computers & Mathematics with Applications 22(8), 91–94 (1991) https://doi.org/10.1016/ 0898-1221(91)90017-X

    Adomian, G.: Solving frontier problems modelled by nonlinear partial differential equations. Computers & Mathematics with Applications 22(8), 91–94 (1991) https://doi.org/10.1016/ 0898-1221(91)90017-X

    work page 1991

Show all 43 references
  1. [10]

    Springer, Dordrecht (1994)

    Adomian, G.: Solving Frontier Problems of Physics: The Decompos ition Method. Springer, Dordrecht (1994). https://doi.org/10.1007/978-94-015-8289-6

    1994 doi

  2. [11]

    The European Physical Journal Plus 132(5), 236 (2017) https://doi.org/10.1140/epjp/i2017-11506- 9

    Mohyud-Din, S.T., Sikander, W., Khan, U., Ahmed, N.: Optimal varia tional iteration method using Adomian’s polynomials for physical problems on finite and semi-infi nite intervals. The European Physical Journal Plus 132(5), 236 (2017) https://doi.org/10.1140/epjp/i2017-11506- 9

    2017 doi

  3. [12]

    The European Physical Journal Plus 137(11), 1291 (2022) https://doi.org/10.1140/epjp/s13360-022-03480-2

    Hamrelaine, S., Kezzar, M., Sari, M.R., Eid, M.R.: Analytical investiga tion of hydromagnetic ferro-nanofluid flowing via rotating convergent/divergent chann els. The European Physical Journal Plus 137(11), 1291 (2022) https://doi.org/10.1140/epjp/s13360-022-03480-2

    2022 doi

  4. [13]

    Journal of Applied Mathematics and Computing 68(3), 2065–2082 (2022) https://doi.org/10.1007/s12190- 021-01613-x

    Panda, A., Santra, S., Mohapatra, J.: Adomian decomposition an d homotopy perturbation method for the solution of time fractional partial integro-differen tial equations. Journal of Applied Mathematics and Computing 68(3), 2065–2082 (2022) https://doi.org/10.1007/s12190- 021-01613-x

    2022 doi

  5. [14]

    The European Physical J ournal Plus 137(1), 63 (2022) https://doi.org/10.1140/epjp/s13360-021-02301-2

    Singh, R., Wazwaz, A.-M.: Analytical approximations of three-po int generalized Thomas–Fermi and Lane–Emden–Fowler type equations. The European Physical J ournal Plus 137(1), 63 (2022) https://doi.org/10.1140/epjp/s13360-021-02301-2

    2022 doi

  6. [15]

    The European Physical Journal Plus 138(3), 300 (2023) https://doi.org/10.1140/epjp/ s13360-023-03905-6

    Naghdi, M.: Solutions for scalar equations in AdS 4 with Adomian method and boundary CFT 3 duals. The European Physical Journal Plus 138(3), 300 (2023) https://doi.org/10.1140/epjp/ s13360-023-03905-6

    2023 doi

  7. [16]

    Numerical Methods for Partial Differential Equatio ns 27(4), 749–766 (2011) https://doi.org/10.1002/num.20549

    Abdelrazec, A., Pelinovsky, D.: Convergence of the Adomian dec omposition method for initial- value problems. Numerical Methods for Partial Differential Equatio ns 27(4), 749–766 (2011) https://doi.org/10.1002/num.20549

    2011 doi

  8. [17]

    Communicat ions in Nonlinear Science and Numerical Simulation 56, 354–364 (2018) https://doi.org/10.1016/j.cnsns.2017.08.025

    Zhang, X., Zou, L., Liang, S., Liu, C.: A novel analytic approximatio n method with a convergence acceleration parameter for solving nonlinear problems. Communicat ions in Nonlinear Science and Numerical Simulation 56, 354–364 (2018) https://doi.org/10.1016/j.cnsns.2017.08.025

    2018 doi

  9. [18]

    Mathematics and Computers in Simulation 40(1), 107–114 (1995) https://doi.org/10.1016/0378-4754(95)00021-8

    Adomian, G.: Solving the mathematical models of neurosciences a nd medicine. Mathematics and Computers in Simulation 40(1), 107–114 (1995) https://doi.org/10.1016/0378-4754(95)00021-8

    1995 doi

  10. [19]

    Mathematical and Computer Modelling 24(11), 39–46 (1996) https://doi.org/10.1016/S0895-7177(96)00171-9

    Adomian, G., Rach, R.: Modified Adomian Polynomials. Mathematical and Computer Modelling 24(11), 39–46 (1996) https://doi.org/10.1016/S0895-7177(96)00171-9

    1996 doi

  11. [20]

    Journal of Applied Mathematics and Physics 04(12), 2215–2232 (2016) https://doi

    Al-Shareef, A., Al Qarni, A.A., Al-Mohalbadi, S., Bakodah, H.O.: Solit on Solutions and Numeri- cal Treatment of the Nonlinear Schrodinger’s Equation Using Modifie d Adomian Decomposition Method. Journal of Applied Mathematics and Physics 04(12), 2215–2232 (2016) https://doi. o...

    2016

  12. [21]

    International Journal of Applied a nd Computational Mathematics 8(2), 81 (2022) https://doi.org/10.1007/s40819-022-01285-6 7

    Kumar, M., Umesh: Recent Development of Adomian Decompositio n Method for Ordinary and Partial Differential Equations. International Journal of Applied a nd Computational Mathematics 8(2), 81 (2022) https://doi.org/10.1007/s40819-022-01285-6 7

    2022 doi

  13. [22]

    Comput- ers & Mathematics with Applications 59(2), 622–628 (2010) https://doi.org/10.1016/j.camwa

    Biazar, J., Gholami Porshokuhi, M., Ghanbari, B.: Extracting a ge neral iterative method from an Adomian decomposition method and comparing it to the variational ite ration method. Comput- ers & Mathematics with Applications 59(2), 622–628 (2010) https://doi.org/10.1016/j.camw...

    2010 doi

  14. [23]

    A pplied Mathematics and Computation 202(1), 113–120 (2008) https://doi.org/10.1016/j.amc.2008.01.027

    Daftardar-Gejji, V., Bhalekar, S.: Solving multi-term linear and n on-linear diffusion–wave equations of fractional order by Adomian decomposition method. A pplied Mathematics and Computation 202(1), 113–120 (2008) https://doi.org/10.1016/j.amc.2008.01.027

    2008 doi

  15. [24]

    In: Wang, D.X

    Sadeghinia, A., Kumar, P.: One Solution of Multi-term Fractional D ifferential Equations by Adomian Decomposition Method: Scientific Explanation. In: Wang, D.X . (ed.) Current Topics on Mathematics and Computer Science Vol. 6, pp. 120–130. Book Publis her International, London, ...

    2021 doi

  16. [25]

    Applied Mathematics Letters 48, 177–179 (2015) https://doi.org/10

    Zhang, X., Liang, S.: Adomian decomposition method is a special ca se of Lyapunov’s artificial small parameter method. Applied Mathematics Letters 48, 177–179 (2015) https://doi.org/10. 1016/j.aml.2015.04.011

    2015

  17. [26]

    Journal of Mathematical Analysis and Applications 91(1), 39–46 (1983) https://doi.org/10.1016/0022-247X(83)90090-2

    Adomian, G., Rach, R.: Inversion of nonlinear stochastic operat ors. Journal of Mathematical Analysis and Applications 91(1), 39–46 (1983) https://doi.org/10.1016/0022-247X(83)90090-2

    1983 doi

  18. [27]

    Journal of Math- ematical Analysis and Applications 102(2), 415–419 (1984) https://doi.org/10.1016/0022- 247X(84)90181-1

    Rach, R.: A convenient computational form for the Adomian poly nomials. Journal of Math- ematical Analysis and Applications 102(2), 415–419 (1984) https://doi.org/10.1016/0022- 247X(84)90181-1

    1984 doi

  19. [28]

    Kybernete s 37(7), 910–955 (2008) https://doi.org/10.1108/03684920810884342

    Rach, R.C.: A new definition of the Adomian polynomials. Kybernete s 37(7), 910–955 (2008) https://doi.org/10.1108/03684920810884342

    2008 doi

  20. [29]

    A Touchs tone Book

    Bell, E.T.: Men of Mathematics, 1st touchstone ed edn. A Touchs tone Book. Simon & Schuster, New York (1986)

    1986

  21. [30]

    6, (1772)

    Lagrange, J.-L.: Essai Sur Le Probl` eme Des Trois Corps (Ess ay on the Essay on the Three-Body Problem) vol. 6, (1772)

  22. [31]

    De L’Imprimerie deCrapelet : Chez J.B.M

    Laplace, P.S.: Trait´ e de M´ ecanique C´ eleste. De L’Imprimerie deCrapelet : Chez J.B.M. Duprat, Paris, France (1798)

  23. [32]

    Astrophysics and Space Scien ce Library, vol

    Poincar´ e, H.: The Three-Body Problem and the Equations of Dynamics: Poincar´ e’s Foundational Work on Dynamical Systems Theory. Astrophysics and Space Scien ce Library, vol. 443. Springer, Cham, Switzerland (1890)

  24. [33]

    Annalen der Physik 384(4), 361–376 (1926) https://doi.org/10.1002/andp.19263840404

    Schr¨ odinger, E.: Quantisierung als Eigenwertproblem. Annalen der Physik 384(4), 361–376 (1926) https://doi.org/10.1002/andp.19263840404

    1926 doi

  25. [34]

    Physical Review 28(6), 1049–1070 (1926) https://doi.org/10.1103/PhysRev.28.1049

    Schr¨ odinger, E.: An Undulatory Theory of the Mechanics of At oms and Molecules. Physical Review 28(6), 1049–1070 (1926) https://doi.org/10.1103/PhysRev.28.1049

    1926 doi

  26. [35]

    Canadian Journal of Physics 33(12), 709–712 (1955) https://doi.org/10.1139/p55-087

    Dirac, P.A.M.: Note on the Use of Non-Orthogonal Wave Function s in Perturbation Calculations. Canadian Journal of Physics 33(12), 709–712 (1955) https://doi.org/10.1139/p55-087

    1955 doi

  27. [36]

    Physical Review 76(6), 769–789 (1949) https://doi.org/10.1103/PhysRev.76.769

    Feynman, R.P.: Space-Time Approach to Quantum Electrodynam ics. Physical Review 76(6), 769–789 (1949) https://doi.org/10.1103/PhysRev.76.769

    1949 doi

  28. [37]

    Applied Mathematics and Computation 111(1), 33–51 (2000) https://doi.org/10.1016/S0096- 3003(99)00063-6

    Wazwaz, A.-M.: A new algorithm for calculating adomian polynomials f or nonlinear operators. Applied Mathematics and Computation 111(1), 33–51 (2000) https://doi.org/10.1016/S0096- 3003(99)00063-6

    2000 doi

  29. [38]

    International Jour nal of Computer Mathematics 93(8), 1299–1319 (2016) https://doi.org/10.1080/00207160.2015.1045421 8

    Fatoorehchi, H., Abolghasemi, H.: Series solution of nonlinear diffe rential equations by a novel extension of the Laplace transform method. International Jour nal of Computer Mathematics 93(8), 1299–1319 (2016) https://doi.org/10.1080/00207160.2015.1045421 8

    2016

  30. [39]

    Applicable Analysis and Discrete Mathematics 10(1), 168–185 (2016) https://doi.org/ 10.2298/AADM160123001K

    Kataria, K.K., Vellaisamy, P.: Simple parametrization methods for g enerating Adomian poly- nomials. Applicable Analysis and Discrete Mathematics 10(1), 168–185 (2016) https://doi.org/ 10.2298/AADM160123001K

    2016 doi

  31. [40]

    Applied Mathematical Modelling 37(8), 6008–6017 (2013) https://doi.org/10.1016/j.apm.2012.12.007

    Fatoorehchi, H., Abolghasemi, H.: Improving the differential tra nsform method: A novel tech- nique to obtain the differential transforms of nonlinearities by the A domian polynomials. Applied Mathematical Modelling 37(8), 6008–6017 (2013) https://doi.org/10.1016/j.apm.2012.12.007

    2013 doi

  32. [41]

    Physical Review E 71(3), 036702 (2005) https://doi.org/10.1103/PhysRevE.71.036702

    Kao, Y.-M., Jiang, T.F.: Adomian’s decomposition method for eigenv alue problems. Physical Review E 71(3), 036702 (2005) https://doi.org/10.1103/PhysRevE.71.036702

    2005 doi

  33. [42]

    Part 2: Numerical application

    Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.-M.: Lyapunov Cha racteristic Exponents for smooth dynamical systems and for hamiltonian systems; A meth od for computing all of them. Part 2: Numerical application. Meccanica 15(1), 21–30 (1980) https://doi.org/10.1007/ BF02128237

    1980

  34. [43]

    Physica D: Nonlinear Phenomena 16(3), 285–317 (1985) https://doi.org/10.1016/ 0167-2789(85)90011-9

    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyap unov exponents from a time series. Physica D: Nonlinear Phenomena 16(3), 285–317 (1985) https://doi.org/10.1016/ 0167-2789(85)90011-9

    1985

  35. [44]

    Kim, A.S.: Complete analytic solutions for convection-diffusion-re action-source equations with- out using an inverse Laplace transform. Scientific Reports 10(1), 8040 (2020) https://doi.org/ 10.1038/s41598-020-63982-w Appendix A Understanding An coefficient In the ADM, the true s...

    2020 doi

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