Singular Initial Value Problem for a System of Integro‐Differential Equations

Abstract

Analytical properties like existence, uniqueness, and asymptotic behavior of solutions are studied for the following singular initial value problem:gi(t)yi′(t)=aiyi(t)(1+fi(t,y(t),∫0+tKi(t,s,y(t),y(s))ds)), yi(0+)=0, t∈(0, t0], wherey=(y1, …, yn), ai>0, i=1, …, nare constants andt0>0. An approach which combines topological method of T. Ważewski and Schauder's fixed point theorem is used. Particular attention is paid to construction of asymptotic expansions of solutions for certain classes of systems of integrodifferential equations in a right-hand neighbourhood of a singular point.

Numerical simulation for fractional order stationary neutron transport equation using Haar wavelet collocation method

“…Integro-differential equations (IDEs) have many applications in different fields of mechanical (Šmarda and Khan, 2012), nuclear engineering, chemistry, astronomy, biology, economics, potential theory and electrostatics. An exact solution of this integro-differential equation was found only in particular cases.…”

Novel Approach for Dealing with Partial Differential Equations with Mixed Derivatives

“…Several methods were proposed, for instance, the Laplace transform method [1][2][3], the Mellin transform method [4], the Fourier transform method [5,6], and the Sumudu transform method [7][8][9] and the Green function method [10] for linear cases. Perturbation method [11], variational iteration method [12][13][14], homotopy decomposition and perturbation method [15][16][17][18], and others were developed for both linear and nonlinear cases.…”

Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations

“…The singular Cauchy problem for first-order differential and integro-differential equations resolved or unresolved with respect to the derivatives of unknowns is fairly well studied see, e.g., 1-16 , but the asymptotic properties of the solutions of such equations are only partially understood. Although the singular Cauchy problems were widely considered by using various methods see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][16][17][18] , the method used here is based on a different approach. In particular, we use a combination of the topological method of T. Ważewski see, e.g., 19, 20 and Schauder's fixed point theorem 21 .…”