Exact Solution to Systems of Linear First-Order Integro-Differential Equations with Multipoint and Integral Conditions

A Unified Formulation of Analytical and Numerical Methods for Solving Linear Fredholm Integral Equations

“…Specifically, we use Lagrange interpolating polynomials of the type (38) to approximate u(x) and put the integral equation in (53) in the symbolic form (41). The kernel K(x, t) = e i(x−t) is continuous on [0, 2π] and separable, and therefore the integral equation in (55) can be written as and write (56) in the form (14). By means of (5), det V = 1 − 2πλ, and hence the integral equation (55) has a unique solution if λ = 1 2π .…”

“…For their solution, a variety of methods have been developed; see [5][6][7][8][9] and others. Well-known classical solution techniques are the Direct Computational Method (DCM), the Degenerate Kernel Methods (DKM), the Quadrature Methods (QM), and the Projection Methods (PM); see, for example, the standard treatises [5][6][7], the traditional articles [10][11][12], and the recent papers [13][14][15][16][17][18][19][20][21]. The DCM has the advantage that it delivers the exact solution in closed form, but its application is limited to special cases where the kernel is separable (degenerate) and the integrals involved can be determined analytically.…”

To the solution of integro-differential equations with nonlocal conditions

“…Loaded differential equations of the form (2.10) with various other types of initial-boundary conditions, as well as the corresponding inverse problems, optimal control problems in various settings have been studied by many authors [4,5,10]. For them, the necessary conditions for the existence and uniqueness of the solution were obtained [7], [16]- [18] and various numerical solution schemes were proposed [3,6,8]. Considering a significant increase in the order of the original problem (2.1) and (2.2), the use of such approaches to the study and solution of practical problems of the form (2.1) and (2.2) and related inverse problems, optimal control problems, causes serious difficulties [4,5,10].…”

“…Tamarkin [9,20]. Later, existence and uniqueness conditions for solutions were obtained for various classes of these problems [12,13,21], and problems is associated with obtaining constructive, necessary and sufficient conditions for the existence and uniqueness of the solution easily verified directly from the data of the problem, and its qualitative properties [7], [16]- [19].…”

A Procedure for Factoring and Solving Nonlocal Boundary Value Problems for a Type of Linear Integro-Differential Equations

“…For instance, the necessity of integral conditions in certain models of epidemics and population growth and the effects when simplifying them are explained in [22]. Nonlocal boundary value problems for integro-differential equations have received much attention recently, see [23][24][25][26][27][28] and the references therein. Factorization methods are very important in constructing exact solutions, but their applicability is confined to certain kinds of operators and moreover they cannot be universal for all problems.…”