Systems of nonlinear Volterra integro-differential equations of arbitrary order

Abstract

In this paper, a new approximate method for solving the system of nonlinear Volterra integro-differential equations of arbitrary (integer and fractional) order is introduced. For this purpose, the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) based on the classical Chebyshev polynomials of the first kind has been introduced that can be used to obtain the solution of the integro-differential equations (IDEs). Also, we construct the fractional derivative operational matrix of order α in the Caputo's definition for GFCFs. This method reduces a system of IDEs by collocation method into a system of algebraic equations. Some examples to illustrate the simplicity and the effectiveness of the propose method have been presented.

On Existence of Sequences of Weak Solutions of Fractional Systems with Lipschitz Nonlinearity

“…The fractionalorder partial differential equations have several applications in many fields such as engineering, biophysics, physics, mechanics, chemistry, and biology (see [1][2][3][4][5][6][7]). More and more efforts have been made in the fractional calculus field especially in FDEs (see, for instance, [2,5,[8][9][10][11][12][13][14][27][28][29][30][31][32][33][34][35][36][37][38][39]). Solution existence for a lot of boundary value problems and several nonlinear elementary problems is studied via a huge number of techniques and nonlinear mathematical tools (see [7,[15][16][17][18][19][20][21][22][23]): the theory of critical point, fixed-point theory, technique of monochromatic iterative, theory degree of coincidence, and the change methods.…”

Multiplicity Solutions of Fractional Impulsive p -Laplacian Systems: New Result

“…In viscoelasticity, electrochemistry, power, porous media, and electromagnetism, for instance, see and the references therein. Many articles have recently investigated the existence of solutions to boundary value problems for FDEs, and we refer the reader to one of them [2,[18][19][20][34][35][36][37][38][39][40][41][42][43][44][45][46] and the references therein. For example, Kamache et al [40] investigated the existence of three solutions for a class of fractional p-Laplacian systems using a variational structure and critical point theory.…”

“…In [36], we investigated the existence of solutions of the periodic boundary value problem for a nonlinear impulsive fractional differential equation with periodic boundary conditions:…”

“…Under such boundary conditions, the presence of solutions for impulsive differential equations with variational structures is determined by variational methods. See, for example, [36] as well as the references therein. Many scholars have recently studied fractional differential equations with impulses using variational methods, fixed point theorems, and critical point theory, due to the rapid growth in the theory of fractional calculus and impulsive differential equations, as well as their broad applications in a variety of fields (see, for example, [35,44] and the references therein for a thorough discussion, as well as the sources therein for more details).…”

Infinite Existence Solutions of Fractional Systems with Lipschitz Nonlinearity

“…FDEs have attracted considerable interest due to their ability to model complex phenomena in several fields of science, engineering, physics, biology, and economics (see [1][2][3][4][5][6][7]). In summary, many improvements have been made in the theory of partial calculus and partial differential equations and partial and ordinary differential equations (see [8][9][10][11][12][13][14][15][16][17][18], [2,5]). Numerous studies have explored the existence and solutions of different nonlinear elementary and boundary value problems through the use of various nonlinear analysis tools and techniques (see, for example, [7,[19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]).…”