A method for constructing exact solutions to some classes of Urysohn-type integral equations (with constant limits of integration) is suggested. It generalizes the method for solving nonlinear integral equations of the second kind with a degenerate kernel. The method is based on the solution of the auxiliary linear equation obtained by discarding the nonlinear terms. Some new solutions to specific nonlinear integral equations of the first and second kind are obtained. The generalization of the method to some nonlinear integro-functional, and integro-differential equations is discussed and illustrative examples are given.
There are a considerable number of methods for finding exact solutions to various classes of linear integral equations (e.g., see [1–20]). Numerous exact solutions obtained with these methods have been collected and systematized in the handbook . Unlike linear equations, only a small number of exact solutions to nonlinear integral equations are known [4, 19, 20].
2. Description of the method for nonlinear integral equations
To make it easier to understand, let us first present the method as applied to constructing exact solutions to nonlinear integral equations. After that, its generalization to more complex equations will be given. Let us consider nonlinear integral equations with constant limits of integration (Urysohn-type equations) of the form
(1)To β = 0 there correspond equations of the first kind and to β = 1, equations of the second kind. In the special case φ1(x) = = φn(x) = 0, equation (1) becomes a linear integral equation. If β = 1 and K(x,t) = 0, it is a nonlinear integral equation with a degenerate kernel. In the latter case, solutions to equation (1) are sought in the form of a finite sum, y(x) = f(x) + ∑ Amφm(x), where the constants Am are determined by solving a system of nonlinear algebraic (transcendental) equations [4, 19, 20]. Consider the auxiliary linear integral equation
(2)which is obtained from (1) by discarding the nonlinear terms. Suppose now that equation (2) can be solved for any function f(x) from some class of functions LF. Let Yf(x) denote a solution to the truncated equation (2). Let φm(x) LF(m = 1,…, n). Then the function
(3)solves the original nonlinear equation (1). The function Yφm(x) in (3) is a solution to the linear equation (2) where f(x) is substituted by φm(x). Inserting (3) into (1), collecting the coefficients of f(x), φ1(x), …, φn(x), and equating the coefficients to zero, one arrives at the following system of nonlinear algebraic (transcendental) equations for the constants Am:
(4)In general, formulas (3)–(4) may determine several (or infinitely many) solutions to the nonlinear integral equation (1). In the special case of a linear equation (1) with Ψm(t,y) = φm(t)y, the algebraic system (4) is linear and determines (in the nondegenerate case) a single solution.
The above method may be used for approximate solution of nonlinear integral equations of the more general case by suitably approximating the nonlinear part of the kernel as Ψ(x,t,y(t)) ≈∑ φm(x)Ψm(t,y(t)).
In the special case of β = 1 and K(x,t) = 0, we have Yf(x) = f(x). Then the above method converts into the known method for nonlinear integral equations of the second kind with constant limits of integration and a degenerate kernel [4, 19, 20].
3. Examples of exact solutions to nonlinear integral equations
Example 1. Consider the integral equation of the first kind with a quadratic nonlinearity
The auxiliary linear integral equation
(6)obtained from (5) by setting φ(x) = 0, is the Fourier sine transform up to a constant factor [6, 14, 20, 21]. Is solution is given by
(7)Solutions of the original nonlinear equation (5) are sought in the form
(8)Substituting (8) into (5) and rearranging he terms, one arrives at a quadratic equation for the coefficients A:
(9)where Here and henceforth, it is assumed that all integrals exist. Different solutions (8) correspond to different roots of the quadratic equation.
Example 2. Consider the nonlinear integral equation of the second kind
1.The solution to the auxiliary linear integral equation of the second kind
is expressed as [7, 20]
(12)Solutions of the original nonlinear equation (10) is sought in the form
(13)where Yφ(x) is defined by (12) with f(x) substituted by φ(x). On inserting (13) into (10), one arrives at the following algebraic (transcendental) equation for the coefficient A:
4. GeneralizationsThe method described above for Urysohn-type integral equations of the first and second kind admits a significant generalization. Consider the abstract nonlinear equation for y = y(x)
(15)where L[y] is a linear operator (integral, functional, differential,1 or any other) and the Im[y] are some nonlinear functionals (i.e., numbers for each y(x) from a given class of functions). Examples of nonlinear functionals: Suppose the truncated linear equation
(18)Formulas (17)–(18) can determine several (or even infinitely many) solutions to equation (15).
5. Examples of exact solutions to nonlinear integro-functional and integro-differential equationsExample 3.
Consider the nonlinear functional-integral equation
(19)where 0 ≤ x ≤ a and 0 ≤ t ≤ a. The auxiliary linear functional equation, obtained from (19) by setting φ(x) = 0, is expressed as
(20)Let us substitute x in (20) by a − x to obtain
(21)Eliminating Y (a - x) from (20)–(21), one finds the solution to functional equation (20):
(22)Following the method described in Section 4, one looks for solutions to the original functional-integral equation (19) in the form where the constant A is determined by the algebraic (transcendental) equation Example 4. Consider the nonlinear integral-functional-differential equation
(23)Note that this equation involves the unknown function with two different arguments, y(xsint) and y(t). The auxiliary linear equation, obtained from (23) by setting φ(x) = 0, has the form and is called Schlömilch’s equation. Its solution is expressed as  Following the method described in Section 4, one looks for solutions to the original integral-functional-differential equation (23) in the form The constant A is determined by the algebraic (transcendental) equation
Example 5. Consider the boundary-value problem for the nonlinear integro-differential equation
(24)with the homogeneous boundary conditions
(25)The solution of the auxiliary linear boundary-value problem is expressed as
(26)Therefore solutions to the original boundary-value problem for the nonlinear integro-differential equation (24) subject to the boundary conditions (25) are sought in the form
(27)The function Yφ(x) is determined by the right-hand side of the first formula in (26), where f(x) is substituted by φ(x). Inserting (27) into (24), one arrives at the following algebraic (transcendental) equation for the coefficient A:
(28)In the special case of f(x) = 0 and φ(x) = 1, one should set Yf(x) = 0 and Yφ(x) = x(x - 1) in (27)–(28).
In the case of nonhomogeneous boundary conditions for integro-differential equations, one should perform the change of variable y(x) = (x) + g(x), where g(x) is any sufficiently smooth function satisfying the boundary conditions, to obtain a problem for (x) with homogeneous boundary conditions. For example, if the integro-differential equation (24) is subjected to the nonhomogeneous boundary conditions y(0) = a and y(1) = b, one could take g(x) = a + (b - a)x.
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