Preface
This section discusses a special technique for power series representation of nonlinear equations, known as the modified decomposition method (MDM for short). Its first concept was proposed by Randolph Rach during his discussions with G. Adomian in 1989 that was crystallized later in a paper published with his colleagues G. Adomian and R.E. Meyers in 1992. That is why this technique is sometimes referred to as the Rach--Adomian--Meyers modified decomposition method. The main idea of the MDM is to decompose the nonlinear term into a power series with the aid of the Adomian polynomials (that are genuine polynomials in this case). Then substitution of the power series solutions along with Adomian decomposition leads to full-history recurrence for determination of its coefficients. This and next two sections illustrate applications of modified decomposition method for solving initial value problems for first and second order differential equations.
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Glossary
List of Publications using the Modified Decomposition Method
The crucial aspect of the Adomian decomposition method (ADM) is the assumption that both, the solution to the problem (for simplicity, we illustrate it on the initial value problem for the first order autonomous differential equation)
Another algorithm was proposed by Tamer A. Abassy:
A natural question is for what sequence of functions { un }n≥0 the Adomian polynomials admit simplification? Looking at the structure of these polynomials, we see that if the elements of the sequence { un } satisfy the group property
- monomials ϕn(x) = xn;
- trigonometric monomials ϕn(x)=ejωx=cos(ωx)+jsin(ωx), where j is a unit vector in positive vertical direction on the complex plane ℂ and ω is a real scalar.
Each of the above two sequences (or their shifted versions) generates either Maclaurin power series (or Taylor) expansion or a Fourier series expansion. Since in this section we are looking for power series, we postpone applications of the latter to the second tutorial.
If we are given a sequence of functions { ϕn(x) }n≥0 that satisfies the group property ϕn(x)ϕm(x)=ϕn+m(x), we can expand the solution into (convergent) series
Subscript[A, 0] = f[Subscript[u, 0]];
Z = Table[0, {i, 1, n}, {j, 1, i}];
Do[Z[[m, 1]] = Subscript[u, m], {m, 1, n}];
For[m = 2, m <= n, m++,
For[k = 2, k <= m, k++,
Z[[m, k]] = Expand[Subscript[u, 1]*Z[[m - 1, k - 1]]];
If[Head[Z[[m, k]]] === Plus, Z[[m, k]] = Map[#/Exponent[#, Subscript[u, 1]]&, Z[[m, k]]], Z[[m, k]] = Map[#/Exponent[#, Subscript[u, 1]]&, Z[[m, k]],{0}]]];
For[k = 2, k <= Floor[m/2], k++, Z[[m, k]] = Z[[m, k]] + (Z[[m - k, k]]/. Subscript[u, s_] -> Subscript[u, s + 1])]];
dir = Table[D[f[Subscript[u, 0]], {Subscript[u, 0], k}], {k, 1, n}];
Do[Subscript[A, m] = Take[dir, m].Z[[m]], {m, 1, n}];
Table[Subscript[A, m], {m, 0, n}]]
Example: We start with a simple power nonlinearity: N[y] = y². Using Mathematica, we can find first few terms without a problem.
In this case, we have the general expression An=∑nk=0ukun−k. Therefore,
ADM | MDM |
---|---|
A0(u0) = u0² | A0(c0) = c0² |
A1(u0, u1) =2 u0 u1 | A1(c0, c1) =2 c0 c1 |
A2(u0, u1, u2) = u1² + 2 u0 u2 | A2(c0, c1, c2) = c1² + 2 c0 c2 |
A3(u0, u1, u2, u3) = 2 u1u2 + 2 u0 u3 | A3(c0, c1, c2, c3) = 2 c1c2 + 2 c0 c3 |
⋮ | ⋮ |
Example: Our next nonlinearity is f(y) = y³. The general formula is also available that represents the general term as a double convolution:
ADM | MDM |
---|---|
A0(u0) = u0³ | A0(c0) = c0³ |
A1(u0, u1) = 3 u0² u1 | A1(c0, c1) =2 c0² c1 |
A2(u0, u1, u2) = 3u0u1² + 3 u0² u2 | A2(c0, c1, c2) = 3c0c1² + 3 c0² c2 |
A3(u0, u1, u2, u3) = 3 u0² u3 + 6 u0 u1 u2 + u1³ | A3(c0, c1, c2, c3) = 3 c0² c3 + 6 c0 c1 c2 + c1³ |
⋮ | ⋮ |
Example: Given the artangent Maclaurin series
Adomian polynomials | General term |
---|---|
A0(c0) = A0( 0 ) = 0² = 0; | A0(c0) = c0² |
A1(0, 1) =2 * 0 * 1 = 0; | A1(c0, c1) =2 c0 c1 |
A2(0, 1, 0) = 1² + 2 * 0 = 1, | A2(c0, c1, c2) = c1² + 2 c0 c2 |
A3(0, 1, 0, 1/3) = 2*1*0 + 2*0*1/3 = 0 | A3(c0, c1, c2, c3) = 2 c1c2 + 2 c0 c3 |
⋮ | ⋮ |
Example: Let us consider the exponential function u=ex=∑n≥0xnn!=∑n≥0cnxn, where cn=1n!,n=0,1,2,…. Raising it to the third power, we get
CoefficientList
Example: Let us consider the reciprocal function f(y) = 1/y. In this case, we actually need to find the inverse of power series. It was done previously using multiplication of two series and solving recurrence relation. Here we show that the required series is easily determined with Adomian's decomposition.
ADM | MDM |
---|---|
A0(u0) = 1/u0 | A0(c0) = 1/c0 |
A1(u0,u1)=−u1u20. | A1(c0,c1)=−c1c20. |
A2(u0,u1,u2)=u21u30−u2u20. | A2(c0,c1,c2)=c21c30−c2c20. |
A3(u0,u1,u2,u3)=−u31u40+2u1u2u30−u3u20. | A3(c0,c1,c2,c3)=−c31c40+2c1u2c30−c3c20. |
A4(u0,u1,u2,u3,u4)=u41u50−3u21u2u40+u22+2u1u3u30−u4u20. | A4(c0,c1,c2,c3,c4)=−c41c50−3c21c2c40+c22+2c1u3c30−c4c20. |
A5(u0,u1,u2,u3,u4,u5)=−u51u60+4u31u2u50−3u1u22+3u21u3u40+2u30(u2u3+u1u4)−u5u20. | A5(c0,c1,c2,c3,c4,c5)=−c51c60+4c31c2c50−3c1c22+3c21c3c40+2c30(c2c3+c1c4)−c5c20. |
⋮ | ⋮ |
Reciprocal of exponential function
Reciprocal coefficients for x | General formula |
---|---|
A0(c0) = 1/c0 = 1 | A0(c0) = 1/c0 |
A1(u0,u1)=−c1c20=−11=−1. | A1(c0,c1)=−c1c20. |
A2(c0,c1,c2)=c21c30−c2c20=1−12=12. | A2(c0,c1,c2)=c21c30−c2c20. |
A3(1,1,1/2,1/3!)=−1+22−13!=−13!. | A3(c0,c1,c2,c3)=−c31c40+2c1u2c30−c3c20. |
A4(1,1,1/2,1/3!,1/4!)=11−32+122+23!−14!=14!. | A4(c0,c1,c2,c3,c4)=−c41c50−3c21c2c40+c22+2c1u3c30−c4c20. |
A5(1,1,12,13!,14!,15!)=−11+42−34−33!+2(1213!+14!)−15!=−15!. | A5(c0,c1,c2,c3,c4,c5)=−c51c60+4c31c2c50−3c1c22+3c21c3c40+2c30(c2c3+c1c4)−c5c20. |
⋮ | ⋮ |
Example: Let us consider the arccotangent function on the interval |x| < 1
Assuming[x > 0, Series[f[x], {x, 0, 9}]]
- Abassy, T.A., Improved Adomian decomposition method, Computers and Mathematics with Applications, 2010, Vol.59, pp. 42--54.
- Gonzalez-Parra, G., Acedo, L., Arenas, A., Accuracy of analytical-numerical solutions of the Michaelis-Menten equation, Computational & Applied Mathematics, 2011, Volume 30, No. 2, pp. 445--461.
- Hasan, Y.Q., Solving first-order ordinary differential equations by Modified Adomian decomposition method, Advances in Intelligent Transportation Systems, Vol. 1, No. 4, 2012
- Duan, J.-S. and Rach, R., New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods, Applied Mathematics and Computation, 2011, Vol. 218 No. 6, pp. 2810-2828.
- Khuri, S.A., On the decomposition method for the approximate solution of nonlinear ordinary differential equations, International Journal of Mathematical Education in Science and Technology, 2001, Vol. 32, No. 4, pp. 525--539.
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