Numerical Methods for a Class of Hybrid Weakly Singular Integro-Differential Equations ()
1. Introduction
The original class of integro-differential equations is from an aeroelasticity pro- blem, in which the mathematical model comprises eight integro-differential equations [1] . In this model, the most determinate equation is a scalar weakly singular integro-differential equation of the first kind. For the current study, a new equation comprising additional derivative terms and integro-differential terms with smooth kernel was used. Under an integrable assumption in previous studies, this new equation can be transformed into a Volterra integral equation of the second kind [2] [3] [4] [5] [6] . The remainder of this paper is organized as follows: Section 2 presents the equations. Section 3 presents the approach to the numerical methods from [7] for the linear cases and the revised version for the nonlinear cases. Section 4 presents the numerical results obtained by the methods in Section 4. Section 5 presents the summary.
2. Problem Description
Consider the class of hybrid weakly singular integro-differential equations
(1)
the initial condition
(2)
where b is a positive constant. The operators D and L are defined as follows:
(3)
(4)
where
. (5)
The weighting kernel g is integrable, positive, nondecreasing, and weakly singular at
. Kernel c is smooth on s. The force
is assumed to be locally integrable for
. Although a more general kernel g is suitable, in this study, emphasis was placed on the Abel type kernel and considers
and
for
. A special value of
corresponds to the original aeroelastic problem. The initial condition
is in
space, which is a weighted
space with weight
. The initial value problems (1) and (2) can be expressed as
(6)
provided that the function
(7)
is absolutely continuous and the function
belongs to
. The corresponding weakly singular Volterra integral equation of the hybrid kind is
for
The proposed algorithms use the separating variables method to directly solve Equations (1) and (2). Without loss of generality, assuming b = 1, the equation is expressed as
(8)
for
, with initial data
(9)
where
is a locally integrable function. By the form of the state in the hybrid integro-differential Equation (8), we obtain the following result:
(10)
and Equation (8) can be rewritten as
(11)
3. Numerical Method
3.1. Linear Problems
The proposed method entails discretizing Equation (1). The space mesh points (corresponding to the s variable) are discretized as
, and a new variable
is defined as follows:
(12)
Equation (12) can then be reformulated as a first-order hyperbolic equation
(13)
with the condition
(14)
Next, assume that the solution to Equations (13) and (14) has the form
(15)
where the basis,
is given by
where
is a piecewise linear function. After substituting the special form of
expressed in Equation (15) into Equations (13) and (14), the governing equations for
can be expressed as follows:
(16)
(17)
where
for
. By defining constants
and
, for
and applying the property of
, Equation (17) can be written as
(18)
Note that Equations (16) and (18) can form a system of first order ordinary differential equations. For time t, the discretization contains
for
Define
for
Assume
for
and
With first term of Equation (18) replaced by the first order finite difference, Equations (16) and (18) can now be expressed as follows:
(19)
for
Furthermore, assume a uniform mesh for both space and time variables, the mesh points are
and
Specifi-
cally,
and
for some positive integers
and
. The corres-
ponding differences are defined as
for the time
variable, and
for the space variable. Thus,
and
for
and
Setting
leads to the relation
for
and
and Equations (19) and
(18) lead to the following system:
(20)
and
(21)
for
After defining the corresponding constants
Equation (21) can be simplified as follows:
(22)
for
The connection of the solution
and
is as follows: Because
, for
and
, and
can be obtained for
in the following
case:
for
(23)
Equations above with the initial condition can be set up as
where the vector
comprises the unknowns
The structure of matrix
is
and that of vector
is
3.2. Nonlinear Problems
The second proposed method contains part of the first method. By assuming
then the property of Equation (13):
for
still holds. The discretized Equation (1) follows the study [7] :
(24)
for
is even.
for
is constant for uniform mesh. By applying the property of
Equation (24) can be written as
(25)
Next, assume that
(26)
where the basis,
are the same as above.
3.2.1. c(s) = 1
Equation (25) becomes
(27)
Setting
and assuming
then
(28)
Similarly, setting
then
(29)
For
and
(30)
For
(31)
then, for
(32)
and for
(33)
By collecting Equations (28), (29), (30), (31), (32), (33) and assuming
for
the system
is constructed, where
and that of vector
is
3.2.2. c(s) = s
In this case, Equation (25) becomes
(34)
Setting
and assuming
then
(35)
Similarly, setting
then
(36)
For
and
(37)
For
(38)
then, for
(39)
and for
(40)
By collecting Equations (35), (36), (37), (38), (39), (40) and assuming
for
the system
is constructed, where
and that of vector
is
4. Numerical Examples
A desktop computer (Intel Pentium 4 microprocessor, 2.80 GHz CPU, and 224 MB RAM) was used for testing the examples.
Example 1.
Example 2.
Example 3.
Example 4.
Example 5.
Example 6.
5. Summary
In this study, we present numerical methods for solving a class of hybrid integro-differential equations; the equation of the first kind originates from an aeroelasticity model. The direct method from previous study provides satisfactory results essentially for the linear cases. For the nonlinear cases, a revised method is proposed to obtain more accurate results.