=Paper=
{{Paper
|id=Vol-1662/mod2
|storemode=property
|title=Application of method of special series with functional arbitrariness in construction of solutions of nonlinear partial differential equations
|pdfUrl=https://ceur-ws.org/Vol-1662/mod2.pdf
|volume=Vol-1662
|authors=Mikhail Filimonov,Adven Masih
}}
==Application of method of special series with functional arbitrariness in construction of solutions of nonlinear partial differential equations==
https://ceur-ws.org/Vol-1662/mod2.pdf
Application of method of special series with functional
arbitrariness in construction of solutions of nonlinear
partial differential equations
Mikhail Filimonov1,2 Adven Masih3
fmy@imm.uran.ru adven795@gmail.com
1 – Krasovskii Institute of Mathematics and Mechanics (Yekaterinburg, Russia)
2 – Ural Federal University (Yekaterinburg, Russia)
3 – Quaid-e-Azam University (Islamabad, Pakistan)
Abstract
In the paper an analytical method for constructing solutions of
nonlinear partial differential equations based on the method of special
series is considered. The essence of this approach is in representation of
solutions of an original equation in the form of series with recurrently
coefficients computed in powers of specially selected functions. Such
functions may depend on several independent variables, and also
contain arbitrary functions that may depend on a smaller number of
independent variables. The paper gives an answer to A.F. Sidorov’s
question about using functional arbitrariness in study of convergence
of the constructed series. There are examples how to prove global
convergence by choosing an arbitrary function.
1 Introduction
The most well known and commonly used methods for representation of solutions of nonlinear partial differential
equations are Taylor series and Fourier series (especially for linear equations of mathematical physics). However,
generally, Taylor series have only local convergence which is often quite slow, and Fourier series when applied to
nonlinear equations lead to finding the coefficients as solutions of infinite systems of nonlinear equations, which
have to be truncated to find an approximate ones for the Fourier coefficients. A.F. Sidorov has formulated the
following requirements for the series used for constructing solutions of nonlinear partial differential equations [1]:
• the series have to converge in sufficiently large domains;
• the series have to converge fast (several first terms of the series have to correctly represent a basic features
of the solution);
• the coefficients of the series have to be computed with using non-expensive and effective algorithms;
• a way of construction of the series has to be applicable to a wide class of equations.
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: A.A. Makhnev, S.F. Pravdin (eds.): Proceedings of the 47th International Youth School-conference “Modern Problems in
Mathematics and its Applications”, Yekaterinburg, Russia, 02-Feb-2016, published at http://ceur-ws.org
245
To ensure the above requirements, it is desirable to get the coefficients of the series not by successive differentiating
(as in the Taylor series) but by integrating some simple recurrent systems of ordinary differential equations. It
is also beneficial that in the case of a nonlinear problem the initial part of the chain of ordinary differential
equations is nonlinear, then it makes possible to pass this short segment catching some basic features of the
nonlinear boundary value problem, and the other terms could be determined from a system of linear differential
equations of simpler structure. The following series allow to satisfy these requirements to some extent.
Picard’s method of iteration for ordinary differential equations leads to construction of solutions in the form
of power series. Investigation of the structure of the resulting successive approximations allowed Cauchy to prove
his theorem for analytic equations. Direct transfer of these approach to partial differential equation is impossible.
A method of special series with recurrently computed coefficients, elaborated to represent solutions of nonlinear
partial differential equations [2, 3, 4] is more close to the idea of representation of solutions of nonlinear ordinary
differential by series in powers of functions which are solutions of other equations. An example of a series with
recurrently computed coefficients used to represent solutions of nonlinear ordinary differential equations is the
result of N.P. Erugin’s [5], concerning reexpansion of solutions of equation
∞
X
ẏ = −y 2 + ak y k , ak = const, y(0) = y0 > 0
k=3
in power of z, which is a solution of “reduced” equation
ż = −z 2 , z(0) = z0 > 0.
In this case, for sufficiently small y0 , z0 a solution of the original equation can be represented as a convergent
series
∞
" !#
X
k
y =z 1+z c+ αk (c)z , c = const
k=1
with recurrently calculated coefficients αk (c).
This approach is most ideologically close to the proposed method. In fact, for a class of nonlinear ordinary
differential equations were constructed series convergent to solutions of these equations. Coefficients of these
series are recurrently computed as solutions of other (perhaps simpler) ordinary differential equations. Solutions
of these simpler equations we shall call basic functions, in accordance with the terminology of the method of
special series. In this example, the basic function is a solution of the “reduced” equation z.
2 The method of special series
Let us consider one of constructions of special series for solving Cauchy problem for nonlinear partial differential
equations of the form
∂mu
∂u
ut = F t, u, , · · · , m , u(0, x) = u0 (x), (1)
∂x ∂x
where F is a polynomial of the unknown function u(t, x) and its derivatives with respect to the space variable.
The solution is represented by the series
∞
X
u(t, x) = un (t)P n (t, x) (2)
n=0
by the powers of basic function P (t, x) satisfying the overdetermined system
Px = A(t, P ), Pt = B(t, P ) (3)
with functions A(t, P ) and B(t, P ) being analytic respect to P and such that A(t, 0)≡0 and B(t, 0)≡0. It was
shown that if the initial conditions are written in the form
∞
X
u0 (x) = u0n P n (0, x), (4)
n=0
246
then substituting series (2) into equation (1), collecting similar terms, and taking into account relations (3), we
obtain the sequence of first-order ordinary differential equations for the coefficients un (t)
0
un = Fn (t, un , . . . , u0 ), un (0) = u0n , n = 1, 2, . . . , (5)
where the right-hand sides Fn include only uj with j ≤ n and the coefficients un may be linearly contained
only. Convergence of some special series in representing solutions of Cauchy problems is proved for different
nonlinear partial differential equations [2, 7, 8]. It is also possible to use this method to solve initial-boundary
value problems with exact satisfaction of zero boundary conditions [9].
As an example, we mention the construction of the series (2) for Korteweg-de
Vries equation with initial condition u(0, x)=u0 (x) in the form of (4). Basic function
1
P (x, t)= , f (t) ∈ C 1 [0, ∞) (6)
exp x+f (t)
is used, which satisfies system (3) for
A(t, P )=−P +f (t)P 2 , B(t, P )=−f (t)0 P 2
with an √ arbitrary function
√ f (t). Convergence of series (2), (6) is proved [2] for all x≥0, t≥0, if
0≤f (t)≤( 3 2−1)/(2− 3 2). At x=0 the arbitrary function f (t) gives rise a new boundary condition u(0, t)=h(t).
We can try to find the function f (t) from the function h(t) [6].
However, such an arbitrary functions can also be used to prove global convergence of special series with
recurrently computed coefficients. Note that condition (3) is not necessary to find the coefficients recurrently.
This recurrency is due to specifics of the studied nonlinear equations [3, 10]. In this case, such special series are
called consistent special series. Further, we consider examples of consistent series.
3 Construction of consistent special series
It is convenient to show construction and application of consistent series on the base of example of model nonlinear
equation
ut = G(t, u, xux , ux /x, u2x ), (7)
where G is a polynomial by u, xux , ux /x, u2x . Consider for this equation Cauchy problem
u(x, 0) = u0 (x), −∞ < x < ∞. (8)
Because the right-hand side of equation (7) contains independent variable x, then its solution cannot be
represented in the form of (2), (3). However, due to specific of the equation, we can find a basis function that
allows the series coefficients to be computed recurently.
Note that equation (7) is reduced to the transport equation by some contact transformation, direct application
of the proposed method to the equation (7) is possible.
Assume that the coefficients of the polynomial G are continuous and limited functions for all t ≥ 0. Solution
of equation (7) will construct in the form of series
∞
X
u(x, t)= un (t)R1n (x, t), (9)
n=0
where
1
R1 (x, t)= 2 , f (t)∈C 1 [0, ∞). (10)
x +f (t)
Let initial condition (8) can be represented as a convergent series
∞
X un0
u0 (x) = 2 + 1)n
, un0 = const. (11)
n=0
(x
We have
Theorem 1. Let the following conditions are satisfied:
247
1) let for constant un0 , defining initial data (11), the following inequalities are valid
Mn
|un0 | ≤ , n ≥ 1, 0 < M < 1,
2n3
2) for an arbitrary function f (t) ∈ C 1 [0, ∞) conditions
f (0) = 1, f (t) ≥ 1, |f 0 (t)| ≤ d1 exp(d2 t), t ≥ 0, d1 , d2 > 0
are valid.
Then series (9), (10) is a formal solution of equation (7) and there exists a number b > 0 such that the series
converges uniformly to the solution of Cauchy problem (7), (8), (11) for all −∞ < x < ∞, 0≤t≤T =b−1 ln M −1 .
Prof. First, we show that the series (9), (10) is a formal solution of the equation (7). Indeed, for a series (9)
the following equalities are valid:
∞
[f (n − 1)un−1 − nun ]R1n ,
P
xux = 2
n=1
∞
ux X
= −2 un−1 (n − 1)R1n ,
x
∞
n=2 (12)
u2x = 4 [m(k − 1)um uk−1 − f (m − 1)(k − 1)um−1 uk−1 ]R1n ,
P P
n=2 m+k=n
∞
[u0n − (n − 1)un−1 R1n ].
P
ut =
n=0
Therefore, by substituting the series (9), (10) into equation (7), taking into account relations (12) for the
coefficients un (t) we obtain a sequence of linear first order differential equations
u0n = Sn (t, f, f 0 , uk (t)), un (0) = un0 , n ≥ 0, k ≤ n. (13)
Consequently, the series (9), (10) is a formal solution of the equation (7).
Using an explicit form of solutions of (13) for the coefficients un (t) the following estimations are proved by
induction:
Mn
|un (t)| ≤ 3 exp(bnt), n ≥ 1, t ≥ 0.
n
With these estimations, we can easily prove convergence of series (9), (10) to solution of Cauchy problem (7),
(8), (11 for all −∞ < x < ∞, 0≤t≤T =b−1 ln M −1 . Theorem 1 is proved.
4 Global convergence of consistent special series with functional arbitrariness
For example of model of nonlinear equations let show that, using the arbitrary function, available in the basic
functions, it is possible to construct a consistent series that converges to a solution of a Cauchy problem for all
−∞ < x < ∞, t ≥ 0.
For equation
ux
ut = + H(t, u, u2x ) (14)
2x
consider a Cauchy problem with initial condition (11).
Let function H can be represented in the form
N1
X X
H(t, u, u2x ) = γmk (t)um u2k
x . (15)
ν=2 m+k=ν
Then equation (14) is a special case of equation (7). Consequently, by Theorem 1, the solution can be found
in the form of (9), (10). After substitution of consistent special series (9), (10) into equation (14), (15), taking
248
into account relations (12), we obtain for the following sequence of linear differential equations to determ the
coefficients un (t):
N1 m
u0n =(f 0 −1)(n−1)un−1 +
P P P P Q
γmk (t) uni
ν=2 m+k=ν j0 +j1 =n n1 +...+nm =j0 i=1
k (16)
4k
P Q P
× [p(q − 1)up uq−1 − f up−1 uq−1 ], n≥1
l1 +...+lk =j1 i=1 p+q=li
with initial data un (0) = un0 .
We show that by choosing an arbitrary function f (t) can be construct a series with recurrently computed
coefficients, converging to the solution for all x and t ≥ 0. We have
Theorem 2. Suppose that the following conditions are valid:
1) γmk (t) ∈ C[0, ∞), |γmk (t)| ≤ γ0 , t ≥ 0, γ0 ≥ 0,
2) u00 = 0, and for the constant un0 , n ≥ 1 the following inequalities are satisfied
M
|un0 | ≤ , M, M0 = const 0 < M ≤ M0 = M0 (γ0 , N1 ),
2n3
3) as a function f (t) we consider
f (t) ≡ 1 + t.
Then consistent special series (9), (10) converges uniformly to the solution of problem (14), (15),(11) for all
−∞ < x < ∞ and t ≥ 0.
Proof. Since f (t) = 1 + t, then the solution of equation (16) may be written in the form
N1
Rt P P P P m
Q
un =un0 + 0 γmk (τ ) uni
ν=2 m+k=ν j0 +j1 =n n1 +...+nm =j0 i=1
k (17)
k
P Q P
× 4 [p(q − 1)up uq−1 − f up−1 uq−1 ]dτ, n ≥ 1.
l1 +...+lk =j1 i=1 p+q=li
By function f (t) it is possible to equal the expression with coefficient un−1 to zero in the right side of equation
(16). I.e., the following equality is valid
(f 0 −1)(n−1)un−1 ≡ 0.
This is one of the essential factors, taking into account the specifics of the equation, and thus it allows to prove
global convergence of the consistent series.
By the method of mathematical induction using direct evaluation of the right side of formula (17), we prove
inequality
M
|un (t)| ≤ 3 (1 + t)n−1 , t ≥ 0, n ≥ 1. (18)
n
If n = 1, u1 (t) ≡ u10 and inequality (18) is valid. Assuming the validity of estimates (18) with n ≤ N , we
carry out the proof for n = N + 1.
To simplify the calculations we carry out the proof of estimates (18) for the following equation:
ux
ut = + u2 , (19)
2x
which is a particular case of equation (14). For n = 2
M M2 M 1 M
|u2 | = |u20 + u210 t| ≤ |u20 | + |u210 t| ≤ + t ≤ + 2M 0 t ≤ 3 (1 + t).
24 4 23 2 2
Here we assume that M0 ≤ 12 . I.e., inequality (18) is true. If n = N + 1 for equation (18) formula (17) has the
form Z t X
uN +1 =uN +1,0 + uk (τ )um (τ )dτ.
0 k+m=N +1
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For |aN +1 | the following inequalities are valid
Z t X
|uN +1 |≤|uN +1,0 |+ |uk (τ )||um (τ )|dτ
0 k+m=N +1
Z t
M X (1 + τ )k−1 (1 + τ )m−1
≤ 3 + M2 dτ
2n 0 k+m=N +1 k3 m3
3 2
M 1
N4 π M
≤ + M 0 (1 + t) ≤ (1 + t)N .
(N + 1)3 2 3 (N + 1)3
Here we assume that M0 ≤ 12 433π2 . Consequently, inequality (18) is valid for any n.
These estimates allow us to prove convergence of series (9), (10)
∞ ∞
M X (1+t)n−1 M X 1 M π2
|u(x, t)|≤ 2 3 2 n−1
≤ = .
x +1+t n=1 n (x +1+t) (x +1+t) n=1 n 6(x2 +1+t)
2 3
Similarly we can prove convergence of the series corresponding ut , ux . Theorem 2 is proved.
Remark 1. Convergence of (9), (10) can be proved under condition that
|f 0 (t) − 1| ≤ M2 = M2 (γ0 , N1 ), M2 = const, t ≥ 0. (20)
Otherwise, the series can diverge.
Let show that if condition (20) is not satisfied, then series (9), (10) diverges when t ≥ 1. Let f (t) ≡ 1, i.e. the
basic function is R1 (x, t) ≡ (x2 + 1)−1 . Consider for linear equation
ux
ut = (21)
2x
a Cauchy problem
M
u(x, 0) = . (22)
x2 + 1
Then the coefficients of series (9), (10) satisfy equations
u0n = −(n − 1)un−1 , n ≥ 1, u1 (0) = M, un (0) = 0
and the solution of problem (21) (22) has the form
∞
X (−1)n+1 tn−1
u(x, t) = M .
n=1
(x2 + 1)n
Thus
∞
X
u(0, t) = M (−1)n+1 tn−1
n=1
and for t ≥ 1 the series diverges. Hence, we have shown that by choosing function f (t), which is increased in a
certain way, we can prove global convergence of series (9), (10). Otherwise, the series can diverge.
Remark 2. Constructing global solutions for a class of nonlinear equations in the form of consistent special
series, not only the form of these equations is important, but also the magnitude of the initial data is important
too.
In the case of initial value problem (22) for linear equation (21) only selected function f (t) is important,
and the value of the constant M could be arbitrary, but for nonlinear equations the value of M is essential in
constructing global solutions.
250
Consider for nonlinear equation (19) initial problem (22). The solution will be constructed in the form of
consistent special series (9), (10) with function f (t) ≡ 1 + t. Then the coefficients un (t) are determined from
equations X
u1 = M, u0n = uk um , un (0) = 0, n ≥ 2
k+m=n
and can be found explicitly
un = M n tn−1 .
The final solution of problem (19), (22) can be represented in the form of consistent series
∞
X M n tn−1
u(x, t) = (23)
n=1
(x2 + 1 + t)n
equals to this sum
M
s(x, t) = . (24)
x2 + 1 + t(1 − M )
Expression (24) is an exact solution of (19), (22) and M > 1 in solution s(x, t) a peculiarity occurs with increasing
t, and series (23) will also diverge. For 0 < M ≤ 1 we have an exact global solution s(x, t) of the problem and
series (23), which converges for all x and t ≥ 0 to this solution.
5 Conclusion
Thus, a positive answer to the A.F. Sidorov’s question about the possibility of using the arbitrary function
is given, This function included to the basic functions of special series is used to prove convergence of the
constructed series to the solution of nonlinear partial differential equations. It was shown that the choice of
an arbitrary function allows to prove convergence of special series to the solution of the equation for all x and
t ≥ 0. In addition, it was found that the magnitude of the initial data is significantly affected on the region of
convergence of special series.
Acknowledgements
The work was supported by Act 211 Government of the Russian Federation, contract 02.A03.21.0006, and by
Russian Foundation for Basic Research 16–01–00401.
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