This section discusses application of the modified decomposition method (MDM for short) to solve the initial value problems for first order nonlinear differential equations. The first concept of it was proposed by Randolf Rach in 1989 that was crystallized later in a paper published with his colleagues G. Adomian and R.E. Meyers. That is way this technique is sometimes referred to as the Rach--Adomian--Meyers modified decomposition method.
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The modified decomposition method (MDM) is
both a decelerated Adomian decomposition method and a developed power series method that nicely and accurately treats any analytic nonlinearity in differential equations. In case of autonomous homogeneous or nonhomogeneous equations, the method is seemed to be superior over other methods. Therefore, we start demonstration of this method in initial value problems for first order differential equations of the form
L[y(t)]=N[y(t)]+g(t),y(t0)=y0,
where L[y(t)]=y′(t)+a(t)y(t)=dy/dt+a(t)y(t) is the linear differential operator, N[y(t)] is a nonlinear term, and g is the system input and y is the system output. The differential operator can be rewritten as L=D+a(t)I, where D=d/dt is the derivative operator and I is the identical operator. Since the unbounded derivative operator has a one-dimensional kernel (the set of functions that are annihilated by the derivative operator), its inverse, calling antiderivative, is a multi-valued operator depending on some constant. Similar property is valid for L. In order to knock out a single inverse operator L-1, we consider L on a space of functions that satisfy an initial condition; more over, it is assumed that an explicit formula for it is available.
Furthermore, we emphasize that the choice for L and concomitantly its
inverse L-1 are determined by the particular equation to be
solved. Hence, the choice for L is nonunique, e.g., for cases of differential
equations with singular coefficients, a different form for the linear
operator L can be chosen. As a rule, the highest derivative is taken
as the linear operator L. Later we consider more general equation including coefficients depending on independent variable.
The Rach--Adomian--Meyers MDM decomposes the solution into a power series
y(t)=∑n≥0cn(t−t0)n
and decomposes the nonlinearity into a power series
according to the Adomian--Rach theorem
where An are the Adomian polynomials in terms of solution coefficients. Of course, the above formula is nothing more than Taylor's series expansion of the nonlinear function N[y]. However, the main difference of the MDM expansion from the power series is that in the former the coefficients depend recursively on the previous coefficients whereas Taylor's coefficients all depend on the infinitesimal behavior of the slope function, but not on previously found coefficients. It is convenient to make a shift in independent variable x = t - t0 to transfer the initial point from t0 into the origin:
y(x)=y0+∑n≥1cnxnandN[y(x)]=∑n≥0An(c0,c1,…,cn)xn.
The Adomian polynomials are evaluated in a similar way as in the ADM
An=1n!∂n∂λn(N[∑n≥0unλn])λ=0,n=0,1,2,…;
but with substitution cn instead of un:
An(u0,u1,…,un)=xnAn(c0,c1,…,cn),n=0,1,2,….
Mathematica code for evaluating Adomian polynomials (Jun-Sheng Duan, An efficient algorithm for the multivariable Adomian polynomials, Applied Mathematics and Computation, 2010, 217, pp. 2456--2467):
We begin with the
Adomian form of the ordinary differential equation, but instead employ the Taylor expansion series for the solution and the
nonlinear function of the solution. The later is possible because of the nonlinear transformation of series by the Adomian–
Rach Theorem
Example:
The Michaelis--Menten equation (MM for short) is employed to describe the kinetics of enzyme-catalysed reactions. Enzymesare proteins that catalyse the chemical reactions essential for living organisms. Therefore, we will solve the MM equation
s′(r)=−Vms(r)Km+s(r),s(0)=a,
where s(r) is the substrate concentration, Vm and Km are the limiting rate and Michaelis constant, respectively. In general,the explicit closed form of the solution is
s(r)=KmW{aKmexp(a−VmrKm)},
where W is the Lambert function and 𝑎 is a constant value from initial condition that satisfies according to the IVP.
We apply the modified decomposition method assuming that the solution is represented by a convergent power series
s(r)=∑n≥0cnrn=a+∑n≥1cnrn,
where the initial term matches the initial condition. The nonlinearity is expanded into the Adomian series
We rewrite the first order equation (1) in explicit way.
Once we expand into Maclauirin power series every term in the given differential equation (assuming that shifting of independent variable was done)
Example:
Consider the first-order nonlinear
differential equation that contains arbitrary holomorphic function
dydx+2xf(y(x))=3x,y(0)=1,
where f is a known function having required number of derivatives (generally speaking, all derivatives). Assuming that the given differential equation has a convergent Maclaurin series solution
y(x)=∑n≥0cnxn=1+∑n≥1cnxn,
we expand the nonlinear term into Adomian's series
f(y(x))=∑n≥0An(c0,c1,…,cn)xn=∑n≥0Anxn.
Taking the derivative, we get
y′(x)=∑n≥0(n+1)cn+1xn.
Similar, we expand the nonlinear term:
xf(y(x))=∑n≥0An(c0,c1,…,cn)xn+1=A0x+∑n≥0An+1xn+2,
where An are Adomian's polynomials for the nonlinearity N[y].
Substituting these identities into the given differential equation, we obtain
c1+2c2x+∑n≥0(n+3)cn+3xn+2=3x−2A0x−2∑n≥0An+1xn+2.
From the initial condition, we find immediately that c0 = 1, and for all other coefficients, we get the recurrence relation by equating the coefficients of like powers of x:
gives a polynomial approximation to the solution.
■
Example:
Consider the following initial value problem for the Riccati differential equation:
dydx=x2+y2(x),y(x0)=y0=1.
Its solution is expressed through Bessel functions:
y(x)={−xJ−3/4(x22)(Γ2(34)+π)−Y−3/4(x22)Γ2(34)(Γ2(34)+π)J1/4(x22)−Y1/4(x22)Γ2(34),x<0,1,x=0,−xJ−3/4(x22)(−Γ2(34)+π)+Y−3/4(x22)Γ2(34)(−Γ2(34)+π)J1/4(x22)+Y1/4(x22)Γ2(34),x>0.
Its series solution can also be found with Mathematica:
y(x)=1+x+x2+34x3+76x4+65x5+3730x6+404315x7+369280x8+428315x9+O(x10).
It is convenient to rewrite the given differential equation in operator form:
L[y]=f(y)+g(x),
where
L[y]=Dy(x)=dydx,g(x)=x2,f(y)=y2(x).
The inverse of the unbounded differential operator L = D dependends on an arbitrary constant, which we denote as c:
L−1[u]=c+∫xx0u(t)dt.
Application of the bounded operator to the given differential equation yields
the functional equation for which the fixed point theorem can be applied:
y(x)=L−1[g(x)]+L−1[f(y)].
The Adomian decomposition method (ADM) states
that the dependent variable y(x) and the nonlinear term
f(y) = y² should be written as the following
infinite series
y(x)=∑n≥0un(x),f(y(x))=∑n≥0An(u0,u1,u2,…,un),
where Ak is the k-th Adomian polynomial. Instead of the standard ADM, we apply the modified decomposition method, which is based on power series expansion
According to the standard ADM, we assume that solution is represented by infinite series
y(x)=u0(x)+∑k≥1uk(x),
where The components un(x) are solutions of the initial value problems
u′n=An−1(u0,u1,…un−1),un(0)=0,n=1,2,…,
and the initial term is determined from the IVP:
u′0=tan(2x)−cos(2x),y(0)=0.
Integrate[Tan[2*s] - Cos[2*s], s]
So
u0(x)=−12ln[cos(2x)]−12sin(2x).
All other components are determined recursively by solving the initial value problems subject that the Adomian polynomials are available. Using the nonlinear term N[y] = ey, we get the first Adomian polynomial directly
However, we are not going to apply the Adomian decomposition method, but its modification. So we are lokking for its solution in terms of power series
y(x)=c0+c1x+c2x2+⋯=∑n≥1cnxn,andy′(x)=∑n≥1ncnxn−1,
where the initial term u0 = 0 is set to zero due to the initial condition y(0) = 0. Next, we expand the input term and the nonlinearity into the power series:
We consider the following initial value problem for the nonlinear differential equation with holomorphic input function g(x) and holomorphic coefficients
dydx=a(x)y(x)+b(x)N[y(x)]+g(x),y(0)=y0.
We assume that this problem has a Maclaurin power series solution
y(x)=c0+∑k≥1ckxk,withc0=y(0)=y0.
Since all known functions in the above differential equation are holomorphic, we have
where An are the Adomian polynomials corresponding to the nonlinearity. Substituting these series into the given differential equation, we obtain the equation between series:
∑k≥1kckxk−1=∑n≥0[n∑k=0an−kck+n∑k=0bn−kAk+gn]xn.
Equating like powers of x, we get a full-history recurrence for unknown coefficients cn:
(n+1)cn+1=n∑k=0an−kck+n∑k=0bn−kAk+gn,n=0,1,2,….
Example:
Consider the first-order nonlinear differential equation with a sinusoidal nonlinearity
dydx=cos(2y−x)+1,y(0)=π/4.
This problem can be solve at least implicitly by substitution: z = 2y - x, so y = (z+x)/2. Then for new dependent variable z, we get a separable equation
z′=2y′−1=2(cosz+1)−1=2cosz+1,z(0)=π/2.
Separating variables and integrating, we obtain
dz2cosz+1=dx⟹2√3arctanh[1√3tanz2]=x+C,
where C is a constant of integration.
Integrate[1/(2*Cos[z] + 1), z]
(2 ArcTanh[Tan[z/2]/Sqrt[3]])/Sqrt[3]
Substitution the initial values into the relation yields
Since the given differential equation is a Bernoulli one, its solution can be represented as the product y(t) = u(t) v(t), where u(t) is a solution of the linear part of the Bernoulli equation
Adomian, G., Rach, R., Transformation of series, Applied Mathematics Letters
Volume 4, Issue 4, 1991, Pages 69-71; https://doi.org/10.1016/0893-9659(91)90058-4
Adomian, G., Rach, R., Nonlinear transformation of series—part II, Computers & Mathematics with Applications
Volume 23, Issue 10, May 1992, Pages 79-83; https://doi.org/10.1016/0898-1221(92)90058-P
Andrianov, I.V., Olevskii, V.I., Tokarzewski, S., A modified Adomian's decomposition method, Journal of Applied Mathematics and Mechanics, 1998, Volume 62, Issue 2, Pages 309--314; https://doi.org/10.1016/S0021-8928(98)00040-9
Duan, J.-S., 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, pp. 2810--2828.
Duan, J.-S., Rach, R., Wazwaz, A.-M., Solution of the model of beam-type
micro- and nano-scale electrostatic actuators by a new modified
Adomian decomposition method for nonlinear boundary value problems,
International Journal of Non-Linear Mechanics, 2013, Volume 49, 2013, Pages
159--169.
Rach, R., Adomian, G., Meyers, R., A modified decomposition method, Computers & Mathematics with Applications, 1992,
Volume 23, Issue 1, January 1992, Pages 17-23; https://doi.org/10.1016/0898-1221(92)90076-T
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