Class Of Integro-differential Equations Research Articles

Analysis of Volterra Integrodifferential Equations with the Fractal-Fractional Differential Operator

In this paper, a class of integrodifferential equations with the Caputo fractal-fractional derivative is considered. We study the exact and numerical solutions of the said problem with a fractal-fractional differential operator. The abovementioned operator is arising widely in the mathematical modeling of various physical and biological problems. In our scheme, first, the integrodifferential equation with the fractal-fractional differential operator is converted to an integrodifferential equation with the Riemann–Liouville differential operator. Additionally, the obtained integrodifferential equation is then converted to the equivalent integrodifferential equation involving the Caputo differential operator. Then, we convert the integrodifferential equation under the Caputo differential operator using the Laplace transform to an algebraic equation in the Laplace space. Finally, we convert the obtained solution from the Laplace space into the real domain. Moreover, we utilize two techniques which include analytic inversion and numerical inversion. For numerical inversion of the Laplace transforms, we have to evaluate five methods. Extensive results are presented. Furthermore, for numerical illustration of the abovementioned methods, we consider three problems to demonstrate our results.

A Special Version of the Collocation Method for One Class of Integro-Differential Equations

A linear integro-differential equation with a singular differential operator in the main part is studied. To find its approximate solution in the space of generalized functions, a special version of the generalized collocation method is proposed and justified

A Special Version of the Collocation Method for One Class of Integro-Differential Equations

A Special Version of the Collocation Method for One Class of Integro-Differential Equations

Approximate properties of stochastic functional differential equations with singular perturbations

This work focuses on stochastic functional differential equations (SFDEs) with wide-band noise perturbation, which is a class of actual physical process and has wide application in the modeling of communication and signal system. Using functional derivatives together with martingale methods and weak convergence techniques, this paper examines the asymptotic properties of the underlying systems as the small parameter tends to zero. Based on the techniques, this paper also establishes the average principle for a class of SFDEs with two-time scales. As a special case, this paper also considers a class of integro-differential equations with wide-band noise perturbation and examines its approximate properties. As an example, a scalar stochastic Lotka–Volterra integro-differential system with wide-band noise perturbation is investigated.

On a Class of Integro-Differential Equations in the Singular Case

On a Class of Integro-Differential Equations in the Singular Case

On a New Class of Singular Integro-differential Equations

In this paper for a new class of model and non-model partial integro-differential equations with singularity in the kernel, we obtained integral representation of family of solutions by aid of arbitrary functions. Such type of integro-differential equations are different from Cauchy-type singular integro-differential equations. Cauchy-type singular integro-differential equations are studied by the methods of the theory of analytic functions. In the process of our research the new types of singular integro-differential operators are introduced and main property of entered operators are learned. It is shown that the solution of studied equation is equivalent to the solution of system of two equations with respect to x and y, one of which is integral equation and the other is integro-differential equation. Further, non-model integro-differential equations are studied by regularization method. This regularization method for non-model equation is based on selecting and analysis of a model part of the equation and reduced to the solution of two second kind Volterra type integral equations with weak singularity in the kernel. It is shown that the presence of a non-model part in the equation does not affect to the general structure of the solutions. From here investigation of the model equations for given class of the integro-differential equations becomes important. In the cases, when the solution of given integro-differential equation depends on any arbitrary functions, a Cauchy type problems are investigated.

On Numerical Solution of One Class of Integro-Differential Equations in a Special Case

A complete theory of solvability of a linear integro-differential equation with a coefficient having power-law zeros is developed. For its approximate solution in the space of generalized functions, special generalized versions of the collocation method based on the use of standard polynomials and cubic splines of minimal defect are proposed and justified. Optimality in the order of accuracy of the method is established.

A Numerical Method for Time-Fractional Reaction-Diffusion and Integro Reaction-Diffusion Equation Based on Quasi-Wavelet

In this research work, we focused on finding the numerical solution of time-fractional reaction-diffusion and another class of integro-differential equation known as the integro reaction-diffusion equation. For this, we developed a numerical scheme with the help of quasi-wavelets. The fractional term in the time direction is approximated by using the Crank–Nicolson scheme. The spatial term and the integral term present in integro reaction-diffusion are discretized and approximated with the help of quasi-wavelets. We study this model with Dirichlet boundary conditions. The discretization of these initial and boundary conditions is done with a different approach by the quasi-wavelet-based numerical method. The validity of this proposed method is tested by taking some numerical examples having an exact analytical solution. The accuracy of this method can be seen by error tables which we have drawn between the exact solution and the approximate solution. The effectiveness and validity can be seen by the graphs of the exact and numerical solutions. We conclude that this method has the desired accuracy and has a distinctive local property.

New Type Super Singular Integro-Differential Equation and Its Conjugate Equation

In this paper for a class of model partial integro-differential equation with super singularity in the kernels, is obtained an integral representation of manifold solutions by arbitrary constants. The conjugate equation for the above-mentioned type of equations is also investigated. Such types of integro-differential equations are different from Cauchy-type singular integro-differential equations. Cauchy-type singular integro-differential equations are studied by the methods of theory of analytical functions. However, the method of analytical functions is not applicable for our case of super singular equations with integrals understanding in Riemann–Stieltjes sense. Here, we have used the method of representation the considering equation as a product of two one-dimensional singular first order integro-differential operators. Further, a complete integro-differential equation and its conjugate equation have been investigated. It is shown that in every cases of characteristic equation roots the homogeneous integro-differential equation can have a nontrivial solutions. Non-model equation is investigated by the regularization method. Regularization of non-model equation is based on selecting a model part of equation. On the basis of the analysis of a model part of equation the solution of non-model equation reduced to the solution of a second kind Volterra integral equations with super singular kernel. It is important to emphasize that in contrast to the usual theory of Volterra integral equations, the studied homogeneous integral equation has nontrivial solutions. It is easy to see that the presence of a non-model part in the equation does not affect to the general structure of the obtained solutions. From here investigation of the model equations for given class of the integro-differential equations becomes important.

Superdiffusive Transport Based on Lévy Walks in a Homogeneous Medium: General and Approximate Self-Similar Solutions

The general and approximate self-similar solutions for the Green’s function have been obtained for a wide class of integrodifferential equations for the two- and three-dimensional (in spatial coordinates) nonstationary superdiffusive transport of a perturbation of a homogeneous medium for a finite fixed velocity of carriers. This problem concerns the transport of resonance radiation in astrophysical gases and plasma, migration of animals, and the transfer of energy of electromagnetic waves in the plasma. The case of a model free path distribution function decreasing by a power law with increasing distance has been considered. Numerical calculations have been performed for two particular types of free path distribution functions, including the case with a Lorentzian shape of wings of the spectral line profile for the emission of photons by atoms or ions. The method developed in [A.B. Kukushkin and P.A. Sdvizhenskii, J. Phys. A: Math. Theor. 49, 255002 (2016)] for an infinite velocity of carriers has been used to construct the approximate self-similar solution. The inclusion of a finite velocity corresponds to the generalization of transport based on Levy flights to transport based on “Levy walks with stops.” The self-similar solution for arbitrary one-dimensional superdiffusive transport obtained in [A.B. Kukushkin and A.A. Kulichenko, Phys. Scripta 94, 115009 (2019)] has been applied to the case of the two- and three-dimensional transports. The accuracy of the self-similar solution has been tested by comparing it to the numerically calculated general solution.

On the Generalization of a Solution for a Class of Integro-Differential Equations with Nonseparated Integral Boundary Conditions

In this paper, the existence and uniqueness results of the generalization nonlinear fractional integro-differential equations with nonseparated type integral boundary conditions are investigated. A natural formula of solutions is derived and some new existence and uniqueness results are obtained under some conditions for this class of problems by using standard fixed point theorems and Leray–Schauder degree theory, which extend and supplement some known results. Some examples are discussed for the illustration of the main work.

On the connection between solutions of initial boundary-value problems for a some class of integro-differential PDE and a linear hyperbolic equation

We consider the second initial boundary-value problem for a certain class of second-order integro-differential PDE with integral operator. The connection of its solution with the solution of the standard second linear initial boundary-value problem for the hyperbolic equation is shown. Thus, the nonlinear problem is reduced to a standard linear problem, whose numerical solution can be obtained, for example, by the Fourier method or Galerkin method. The article provides examples of five integro-differential equations for various integral operators as particular representatives of the class of integro-differential equations for a better understanding of the problem. The application of the main theorem to these examples is shown. Some simple natural requirement is imposed on the integral operator; so, in four cases out of five the problem’s solution satisfies some phase constraint. The form of these constraints is of particular interest for the further research.

A Local Galerkin Integral Equation Method for Solving Integro-differential Equations Arising in Oscillating Magnetic Fields

The current work investigates a computational method to solve a class of integro-differential equations which describes the charged particle motion for certain configurations of oscillating magnetic fields. To handle the method, we first convert these types of integro-differential equations to two-dimensional Volterra integral equations of the second kind. Afterward, the solution of the mentioned Volterra integral equations can be estimated using the Galerkin method based on the moving least squares (MLS) approach constructed on scattered points. The MLS methodology is an effective technique for approximating an unknown function that involves a locally weighted least squares polynomial fitting. Since the scheme does not need any background meshes, it can be identified as a meshless method. The scheme is simple and effective to solve integro-differential equations and its algorithm can be easily implemented. Finally, numerical examples are included to show the validity and efficiency of the new technique.

On Fourier – Kontorovich–Lebedev generalized convolution transforms

ABSTRACTWe deal with several classes of integral transformations of the formwhere D is a special differential operator. In case D is the identical operator, we prove a Young type theorem for a generalized convolution operator related to the Fourier and the Kontorovich–Lebedev integral transforms. In case D is a certain finite order differential operator, we study the unitary property of these transforms and apply them to solve a class of integro-differential equations. Several classes of system of integral equations are also considered.

Certain Nonlinear Integral Inequalities and Their Applications

Several new retarded integral inequalities of Gronwall-Bellman-Pachpatte type are presented which generalize the inequalities to the more general nonlinear case in the literature and provide explicit bounds on unknown functions. These results include many existing ones as special cases and can be used as tools in the qualitative analysis of certain classes of integrodifferential equations.

Different types of Hyers-Ulam-Rassias stabilities for a class of integro-differential equations

We study different kinds of stabilities for a class of very general nonlinear integro-differential equations involving a function which depends on the solutions of the integro-differential equations and on an integral of Volterra type. In particular, we will introduce the notion of semi-Hyers-Ulam-Rassias stability, which is a type of stability somehow in-between the Hyers-Ulam and Hyers-Ulam-Rassias stabilities. This is considered in a framework of appropriate metric spaces in which sufficient conditions are obtained in view to guarantee Hyers-Ulam-Rassias, semi-Hyers-Ulam-Rassias and Hyers-Ulam stabilities for such a class of integro-differential equations. We will consider the different situations of having the integrals defined on finite and infinite intervals. Among the used techniques, we have fixed point arguments and generalizations of the Bielecki metric. Examples of the application of the proposed theory are included.

On a New Class of Abstract Neutral Integro-Differential Equations and Applications

In this paper we introduce a new class of abstract integro-differential equations with delay and we study the existence of strict solutions. An application involving the heat equation with memory is presented.

The time discretization in classes of integro-differential equations with completely monotonic kernels: Weighted asymptotic convergence

We study discretization in classes of integro-differential equations $$\begin{gathered} u'(t) + \int_0^t {(\lambda _1 a_1 (t - \tau ) + \lambda _2 a_2 (t - \tau ) + \cdots + \lambda _n a_n (t - \tau ))} u(\tau )d\tau = 0,t > 0, \hfill u(0) = 1,\lambda _j \geqslant 1,j = 1,2,...,n, \hfill \end{gathered} $$ , where the functions aj (t), 1 ⩽ j ⩽ n, are completely monotonic on (0,∞) and locally integrable, but not constant. The equations are discretized using the backward Euler method in combination with order one convolution quadrature for the memory term. The stability properties of the discretization are derived in the weighted l1(ρ; 0,∞) norm, where ρ is a given weight function. Applications to the weighted l1 stability of the numerical solutions of a related equation in Hilbert space are given.

Weighted pseudo almost periodic solutions to a class of semilinear integro-differential equations in Banach spaces

In this paper we prove the existence of weighted pseudo almost periodic mild solutions for the class of integro-differential equations in the form $u'(t)=Au(t)+\alpha\int_{-\infty}^{t}e^{-\beta(t-s)}Au(s) \,ds+f(t,u(t)) $ where $f(\cdot,u(\cdot))$ is a Stepanov-like weighted pseudo almost periodic function and A generates an immediately norm continuous $C_{0}$ -semigroup on a Banach space X. Also, we give a short proof to show that the vector-valued space of Stepanov-like weighted pseudo almost periodic functions is a Banach space.

On a new class of integro-differential equations

We consider various initial-value problems for ordinary integro-differential equations of first order that are characterized by convolution-terms, where all factors depend on the solutions of the equations. Applications of such problems are descriptions of certain glass-transition phenomena based on mode-coupling theory, for instance. We will prove results concerning well-posedness of such problems and the asymptotic behaviour of their solutions.