Integro-differential Equations Of Convolution Type Research Articles

Fractional Linear Volterra Integro-Differential Equations in Banach Spaces

The paper presents the foundations of the theory of linear fractional Volterra integro-differential equations of convolution type in Banach spaces. It is established that the existence of a fractional resolvent operator for such equations is equivalent to the well-posedness of the formulation of the initial problem for them. Within the framework of this approach, a theorem of the Hille–Yosida type is proved.

On the Exact Solution of Nonlocal Euler–Bernoulli Beam Equations via a Direct Approach for Volterra-Fredholm Integro-Differential Equations

First, we develop a direct operator method for solving boundary value problems for a class of nth order linear Volterra–Fredholm integro-differential equations of convolution type. The proposed technique is based on the assumption that the Volterra integro-differential operator is bijective and its inverse is known in closed form. Existence and uniqueness criteria are established and the exact solution is derived. We then apply this method to construct the closed form solution of the fourth order equilibrium equations for the bending of Euler–Bernoulli beams in the context of Eringen’s nonlocal theory of elasticity (two phase integral model) under a transverse distributed load and simply supported boundary conditions. An easy to use algorithm for obtaining the exact solution in a symbolic algebra system is also given.

Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams

Recent developments have shown that the widely used simplified differential model of Eringen’s nonlocal elasticity in nanobeam analysis is not equivalent to the corresponding and initially proposed integral models, the pure integral model and the two-phase integral model, in all cases of loading and boundary conditions. This has resolved a paradox with solutions that are not in line with the expected softening effect of the nonlocal theory that appears in all other cases. In addition, it revived interest in the integral model and the two-phase integral model, which were not used due to their complexity in solving the relevant integral and integro-differential equations, respectively. In this article, we use a direct operator method for solving boundary value problems for nth order linear Volterra–Fredholm integro-differential equations of convolution type to construct closed-form solutions to the two-phase integral model of Euler–Bernoulli nanobeams in bending under transverse distributed load and various types of boundary conditions.

Convolutional Integro-Differential Equations in Banach Spaces With a Noetherian Operator in the Main Part

An initial-value problem for an integro-differential equation of convolution type with a finite index operator for the higher order derivative in Banach spaces is considered. The equations under consideration model the evolution of the processes with "memory" when the current state of the system is influenced not only by the entire history of observations but also by the factors that have formed it and that remain relevant to the current moment of observation. Solutions are constructed in the class of generalized functions with a left bounded support with the use of the theory of fundamental operator functions of degenerate integro-differential operators in Banach spaces. A fundamental operator function that corresponds to the equation under consideration is constructed. Using this function the generalized solution is restored. The relationship between the generalized solution and the classical solution of the original initial-value problem is studied. Two examples of initial-boundary value problems for the integro-differential equations with partial derivatives are considered

Method of Maximal Monotonic Operators in the Theory of Nonlinear Integro-Differential Equations of Convolution Type

Using the method of maximal monotonic (in the Browder–Minty sense) operators, we prove global theorems on the existence and uniqueness of solutions for various classes of nonlinear integrodifferential equations of convolution type in real spaces Lp, 1 < p < ∞, and present illustrative examples.

Finding the Zeros of a High-Degree Polynomial Sequence

A 1-parameter initial-boundary value problem for a linear spatially 1-dimensional homogeneous degenerate wave equation, posed in a space-time rectangle, in case of strong degeneracy, was reduced to a linear integro-differential equation of convolution type (JODEA, 29(1) (2021), pp. 1–31). The former was then solved by applying the Laplace transformation, and the solution formula was inverted in the form of the Neumann series. The current study deals with an other approach to the inversion of the solution formula, based on invoking the Bromwich integral and the Cauchy residue theorem for the integrand. The denominator of the integrand being an infinite series with respect to rational functions of the complex variable, converges quite rapidly and can be approximated with finite series of m terms. Therefore finding the zeros of the approximated denominator reduces to finding the zeroes of a polynomial of degree 2m. For the resulting polynomial sequence some numerical approaches have been applied.

System of integro-differential equations of convolution type with power nonlinearity

System of integro-differential equations of convolution type with power nonlinearity

A System of Integro-Differential Equations of Convolution Type with Power Nonlinearity

A System of Integro-Differential Equations of Convolution Type with Power Nonlinearity

The Modified Sadik Decomposition Method to Solve a System of Nonlinear Fractional Volterra Integro-Differential Equations of Convolution type

In this work, an incorporated form of Sadik transform and Adomian decomposition method which is called the Sadik decomposition method is presented. The method is applied to solve a system of nonlinear fractional Volterra integro-differential equations in the convolution form. To avoid collecting the noise terms that lead the method to fail for seeking the solution, the proposed method is modified by selecting a suitable initial solution. The obtained results are expressed in the explicit form of a power series with easily computable terms. In addition, illustrative examples are shown to demonstrate the effectiveness of the method.

Solutions to a Simplified Initial Boundary Value Problem for 1D Hyperbolic Equation with Interior Degeneracy

A 1-parameter initial boundary value problem (IBVP) for a linear homogeneous degenerate wave equation (JODEA, 28(1), 1) in a space-time rectangle is considered. The origin of degeneracy is the power law coefficient function with respect to the spatial distance to the symmetry line of the rectangle, the exponent being the only parameter of the problem, ranging in (0,1) and (1,2) and producing the weak and strong degeneracy respectively. In the case of weak degeneracy separation of variables is used in the rectangle to obtain the unique bounded continuous solution to the IBVP, having the continuous flux. In the case of strong degeneracy the IBVP splits into the two derived IBVPs posed respectively in left and right half-rectangles and solved separately using separation of variables. Continuous matching of the obtained left and right families of bounded solutions to the IBVPs results in a linear integro-differential equation of convolution type. The Laplace transformation is used to solve the equation and obtain a family of bounded solutions to the IBVP, having the continuous flux and depending on one undetermined function.

To the theory of solvability of degenerate integro-differential equations in Banach spaces

The paper considers a new approach to constructing generalized solutions of degenerate integro-differential equations of convolution type in Banach spaces. The principal idea of the method proposed implies the refusal of the condition of existence of the full Jordan set for the Fredholm operator of the higher derivative with respect to the operator bundle formed by the rest of operator coefficients of the differential part and by the operator kernel of the integral component of the equation. The conditions are superimposed upon the values of the operator function specially constructed on the basis elements of the Fredholm operator kernel. Under such an approach, the differential part of the equation may include not only the higher derivative but also any combination of lower derivatives, what allows one to consider the convolution integral-differential equations from universal positions, without any special account of the structure of the operator bundle. The method proposed represents a form of generalization of the technique based on the application of Jordan sets of Fredholm operators, and, in the case of existence of the latter, the method coincides with this technique. A generalized solution is constructed in the form of a convolution of the fundamental operator function, which corresponds to the equation under investigation, and the function, which includes both the right-hand side of the equation and the initial data. The conditions, under which such a generalized solution does not contain any singular component, and the regular component converts the initial equation into an identity and satisfies the initial data will provide for the resolvability of the initial problem in the class of functions characterized by the respective smoothness. In this case, the generalized solution constructed will be classical. The theorem on the form of fundamental operator function has been proved. The abstract results have been illustrated via examples of initial-boundary value problems of applied character (from the theory electromagnetic fields, the theory of oscillations in visco-elastic media, the theory of vibrations of thermal-elastic plates).

Stability and Accuracy Analysis of $$\theta $$ Scheme for a Convolutional Integro-Differential Equation

Spatially long-range interactions for linearly elastic media resulting in dispersion relations are modelled by an integro-differential equation of convolution type (IDE) that incorporate non-local effects. This type of IDE is nonstandard (hence, it is almost impossible to obtain exact solutions) and plays an important role in modeling various applied science and engineering problems. In this article, such an IDE describing a linear elastic wave phenomenon has been studied. First, a discrete equivalent of the model IDE in space is proposed and then a class of forward backward average one step \(\theta \) scheme for the semi-discrete time dependent numerical method has been developed. Further, stability and accuracy of the developed method has been analyzed rigorously.

An ultraspherical spectral method for linear Fredholm and Volterra integro-differential equations of convolution type

Abstract The Legendre-based ultraspherical spectral method for ordinary differential equations (Olver, S. &amp; Townsend, A. (2013) A fast and well-conditioned spectral method. SIAM Rev., 55, 462–489.) is combined with a formula for the convolution of two Legendre series (Hale, N. &amp; Townsend, A. (2014a) An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput., 36, A1207–A1220.) to produce a new technique for solving linear Fredholm and Volterra integro-differential equations with convolution-type kernels. When the kernel and coefficient functions are sufficiently smooth, then the method is spectrally accurate and the resulting almost-banded linear systems can be solved with linear complexity.

A Non-Markovian Phase Space Approach to Schrödinger Dynamics: The Space-Time Wigner Transform

A phase space description of Schroźdinger dynamics is provided in terms of a quantum kinetic formalism relying on the introduction of an appropriate extension of the well-known Wigner transform, also accounting for time delocalizations. This “space-time Wigner distribution,” built up in the framework of two-time correlation functions, is shown to be governed by a non-Markovian, integro-differential equation of convolution type. Its utility in investigating long time dynamics of quantum systems is also discussed and illustrated with some examples.

An Approximated Solutions for nth Order Linear Delay Integro-Differential Equations of Convolution Type Using B-Spline Functions and Weddle Method

The paper is devoted to solve nth order linear delay integro-differential equations of convolution type (DIDE's-CT) using collocation method with the aid of B-spline functions. A new algorithm with the aid of Matlab language is derived to treat numerically three types (retarded, neutral and mixed) of nth order linear DIDE's-CT using B-spline functions and Weddle rule for calculating the required integrals for these equations. Comparison between approximated and exact results has been given in test examples with suitable graphing for every example for solving three types of linear DIDE's-CT of different orders for conciliated the accuracy of the results of the proposed method.

An Approximated Solutions for nth Order Linear Delay Integro-Differential Equations of Convolution Type Using B-Spline Functions and Weddle Method

The paper is devoted to solve nth order linear delay integro-differential equations of convolution type (DIDE's-CT) using collocation method with the aid of B-spline functions. A new algorithm with the aid of Matlab language is derived to treat numerically three types (retarded, neutral and mixed) of nth order linear DIDE's-CT using B-spline functions and Weddle rule for calculating the required integrals for these equations. Comparison between approximated and exact results has been given in test examples with suitable graphing for every example for solving three types of linear DIDE's-CT of different orders for conciliated the accuracy of the results of the proposed method.

On optimal boundary and distributed control of partial integro–differential equations

Abstract A method of optimal control problems investigation for linear partial integro-differential equations of convolution type is proposed, when control process is carried out by boundary functions and right hand side of equation. Using Fourier real generalized integral transform control problem solution is reduced to minimization procedure of chosen optimality criterion under constraints of equality type on desired control function. Optimality of control impacts is obtained for two criteria, evaluating their linear momentum and total energy. Necessary and sufficient conditions of control problem solvability are obtained for both criteria. Numerical calculations are done and control functions are plotted for both cases of control process realization.

On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems

This paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the Salamon-Weiss sens. We first present results on the validity of the inverse formula of the Laplace transform for the resolvent families associated with scalar Volterra integral equations of convolution type in Banach spaces, which extends and improves the results in Hille and Philllips (1957) and Cioranescu and Lizama (2003, Lemma 5), respectively, including the stronger version for a class of scalar Volterra integrodifferential equations of convolution type on unconditional martingale differences UMD spaces, provided that the leading operator generates aC0-semigroup. Next, a necessary and sufficient condition forLp-admissibilityp∈1,∞of the system's control operator is given in terms of the UMD-property of its underlying control space for a wider class of Volterra integrodifferential equations when the leading operator is not necessarily a generator, which provides a generalization of a result known to hold for the standard Cauchy problem (Bounit et al., 2010, Proposition 3.2).

A direct method to solve integral and integro-differential equations of convolution type by using improved operational matrix

In this article, we use improved operational matrix of block pulse functions on interval [0, 1) to solve Volterra integral and integro-differential equations of convolution type without solving any system and projection method. We first obtain Laplace transform of the problem and then we find numerical inversion of Laplace transform by improved operational matrix of integration. Numerical examples show that the approximate solutions have a good degree of accuracy.

Parallel computations and numerical simulations for nonlinear systems of Volterra integro-differential equations

We investigate thalamo-cortical systems that are modeled by nonlinear Volterra integro-differential equations of convolution type. We divide the systems into smaller subsystems in such a way that each of them is solved separately by a processor working independently of other processors results of which are shared only once in the process of computations. We solve the subsystems concurrently in a parallel computing environment and present results of numerical experiments, which show savings in the run time and therefore efficiency of our approach. For our numerical simulations, we apply different numbers np of processors and each case shows that the run time decreases with increasing np. The optimal speed-up is obtained with np=N, where N is the (moderate) number of equations in the thalamo-cortical model.