Singular Integro-differential Equations Research Articles

New algorithm on linearization-discretization solving systems of nonlinear integro-differential Fredholm equations

This article deals with a new strategy for solving a certain type of nonlinear integro-differential Fredholm equations with a weakly singular kernel. We build our new algorithm starting with the linearization phase using Newton's iterative process, then with the discretization phase we apply the Kantorovich's projection method. The discretized linear scheme will be approximated by the product integration method in the weak singular terms, and the other regular integrals will be approximated by the Nyström method. The process of convergence of our new algorithm is carried out under certain predefined and necessary conditions. Finally, we give practical examples where, the results show the efficiency of our new algorithm for solving systems of weakly singular nonlinear integro-differential equations.

A high-order compact ADI finite difference scheme on uniform meshes for a weakly singular integro-differential equation in three space dimensions

A high-order compact ADI finite difference scheme on uniform meshes for a weakly singular integro-differential equation in three space dimensions

Efficient method for solving nonlinear weakly singular kernel fractional integro-differential equations

<abstract><p>This paper introduced an efficient method to obtain the solution of linear and nonlinear weakly singular kernel fractional integro-differential equations (WSKFIDEs). It used Riemann-Liouville fractional integration (R-LFI) to remove singularities and approximated the regularized problem with a combined approach using the generalized fractional step-Mittag-Leffler function (GFSMLF) and operational integral fractional Mittag matrix (OIFMM) method. The resulting algebraic equations were turned into an optimization problem. We also proved the method's accuracy in approximating any function, as well as its fractional differentiation and integration within WSKFIDEs. The proposed method was performed on some attractive examples in order to show how their solutions behave at various values of the fractional order $ \digamma $. The paper provided a valuable contribution to the field of fractional calculus (FC) by presenting a novel method for solving WSKFIDEs. Additionally, the accuracy of this method was verified by comparing its results with those obtained using other methods.</p></abstract>

Some Singular Integro-Differential Equations

Some Singular Integro-Differential Equations

An efficient parametric finite difference and orthogonal spline approximation for solving the weakly singular nonlinear time-fractional partial integro-differential equation

An efficient parametric finite difference and orthogonal spline approximation for solving the weakly singular nonlinear time-fractional partial integro-differential equation

The Müntz–Legendre Wavelet Collocation Method for Solving Weakly Singular Integro-Differential Equations with Fractional Derivatives

We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some numerical simulations demonstrate the efficiency and accuracy of the method. According to the obtained results, the presented method solves this type of equation well and gives significant results.

Pontryagin’s maximum principle for a fractional integro-differential Lagrange problem

In this paper, we study an optimal control problem of Lagrange type in which a control system is described by a nonlinear and singular integro-differential equation of Volterra type with a Caputo derivative. The necessary first-order optimality conditions for a local in (y,u,v) solution (in the form of a maximum principle) for the considered problem are derived. Our approach to deriving these conditions is based on an extremum principle for an abstract optimal control problem obtained in Idczak and Walczak (2020), where the main assumption is smoothness in (y,u,v) of a cost and an operator describing constraints.

Numerical schemes for a class of singular fractional integro-differential equations

We consider a class of fractional integro-differential equations with certain type of singularities at the origin. Using a change of variables we reformulate the original problem as a cordial Volterra integral equation and study the existence, uniqueness and regularity of the exact solution. With the help of the obtained information we construct a collocation based numerical method for finding the approximate solution of the original problem and analyse the convergence and the convergence order of the proposed method. We also give some numerical examples.

A block-by-block strategy for fractional systems of nonlinear weakly singular integro-differential equations

A block-by-block strategy for fractional systems of nonlinear weakly singular integro-differential equations

Thermal Stressed State of a Bimaterial with Interface Crack Filled with a Compressible Liquid

We formulate the problem of thermoelasticity for a bimaterial with interface crack filled with a liquid with regard for the pressure and heat conductivity of the liquid. We consider the case of partial crack closure on the edges under the action of a compressive load and a heat flux. The problem is reduced to a system of nonlinear singular integrodifferential equations for a temperature jump between the crack faces and its opening displacement. The length of the contact zone is determined from the condition of smooth closure of crack faces at the points separating the open and closed parts of the crack. We analyze the changes in the crack length and height and in the pressure of liquid depending on the compressive loads, density of the heat flux, and its direction. We also determine the dependence of the mode-II stress intensity factor on the level of pressure and the thermal conductivity coefficient of liquid.

On Approximate Solution of One Class of Singular Integro-Differential Equations

On Approximate Solution of One Class of Singular Integro-Differential Equations

A Lagrange spectral collocation method for weakly singular fuzzy fractional Volterra integro-differential equations

A Lagrange spectral collocation method for weakly singular fuzzy fractional Volterra integro-differential equations

Вплив поздовжньої і поперечної теплопровідності міжфазних щілин на ефективні параметри біматеріалу

The effective parameters of the bi-material with a periodic system of interfacial cracks are studied, taking into account their longitudinal and transverse thermal conductivity. The bi-material is subjected to tensile forces and uniform heat flow. The transverse and longitudinal thermal conductivity of the cracks is taken into account by the thermal resistance of the filler and the thermal conductivity of the surface films, respectively. The thermal resistance of the filler is directly proportional to the opening of the cracks and inversely proportional to the thermal conductivity of the filler. Thermal conductivity of surface films does not change under the influence of load. The thermo-elastic problem is reduced to nonlinear systems of singular integro-differential equations for an opening cracks and a temperature jump between the cracks faces. An analytical-numerical iterative procedure for solving this system is proposed. Based on the obtained solution, the effective temperature jump and the effective thermal resistance of the bi-material are determined. The dependences of the effective parameters of the bi-material on the applied load and thermal conductivity coefficients of the filler and the surface films of the cracks are analyzed.

Statement and general technique for solving problem of oscillation of a hereditarily deformable aircraft

This article provides a general statement and a technique for solving the problem of oscillations of a hereditarily deformable aircraft in a gas flow with a finite number of degrees of freedom. Using the Lagrange equations and the variational principle of the hereditary theory of viscoelasticity, the equations of motion of the problem under consideration are derived. The generalized forces acting on the aircraft in the subsonic flight mode are determined according to the stationarity hypothesis. As a result, closed interconnected weakly singular integro-differential equations are obtained that describe the mathematical model of the problem with a finite number of degrees of freedom. General schemes for the numerical solution of these equations are outlined. As an example, the flexural-torsional-aileron flutter of the transient process of a hereditarily deformable wing with a finite freedom number is considered.

Transverse shear of an isotropic elastic medium in the case when the binder and inclusions are weakened by crack initiation

An elastic medium is considered, weakened by a doubly periodic system of round holes, filled with absolutely rigid inclusions, soldered along the bypass and has a crack initiation. The medium (binder) is weakened by two periodic systems of rectilinear crack initiation directed collinear to the abscissa and ordinate axes, and their sizes are not the same. General representations are constructed that describe a class of problems with a doubly periodic stress distribution outside circular holes and cracks under transverse shear. The analysis of the limiting equilibrium of cracks in the framework of the end zone model is carried out on the basis of a nonlocal fracture criterion with a force condition for the propagation of the crack tip and a deformation condition for determining the advancement of the edge of the end zone of the crack. Basic resolving equations are obtained in the form of infinite algebraic systems and three nonlinear singular integro-differential equations. The equations in each approximation were solved by the Gaussian method with the choice of the principal element for different values of the order of M, depending on the radius of the holes. Calculations were carried out to determine the forces in the connections of the end zones and the ultimate loads causing the growth of cracks.

INVESTIGATION OF THE STRESS-STRAIN STATE OF RODS WITH TRANSVERSE CRACKS UNG UNDER UNEVEN HEATING

The problem of fracture mechanics for a strip (rod) weakened by a straight-through crack with end zones when the strip is under the influence of an uneven temperature field is considered. The determination of the forces in the connections between the sides is reduced to the solution of a singular integro-differential equation.

A spectral approach to non-linear weakly singular fractional integro-differential equations

In this work, a class of non-linear weakly singular fractional integro-differential equations is considered, and we first prove existence, uniqueness, and smoothness properties of the solution under certain assumptions on the given data. We propose a numerical method based on spectral Petrov-Galerkin method that handling to the non-smooth behavior of the solution. The most outstanding feature of our approach is to evaluate the approximate solution by means of recurrence relations despite solving complex non-linear algebraic system. Furthermore, the well-known exponential accuracy is established in \(L^{2}\)-norm, and we provide some examples to illustrate the theoretical results and the performance of the proposed method.

The adhesive contact problem for a piecewise-homogeneous orthotropic plate with an elastic patch

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite patch of the wedge-shaped, which meets the interface at a right angle and is loaded with tangential and normal forces is considered. Using methods of the theory of analytic functions, the problem is reduced to the system of singular integro-differential equations (SIDE) with fixed singularity. Under tension-compression of patch using an integral transformation a Riemann problem is obtained, the solution of which is presented in explicit form. The tangential contact stresses along the contact line are determined and their asymptotic behavior in the neighborhood of singular points is established.

Computational technique for multi-dimensional non-linear weakly singular fractional integro-differential equation

In this article, our aim is to construct two new schemes for numerically solving the nonlinear weakly singular integro-fractional differential equation (WSIFDE) in 1D and 2D. Scheme-I uses 1D and 2D shifted Legendre polynomials (SLP) as basis functions whereas scheme-II uses 1D and 2D interpolating basis function (IBF) as basis functions. The main characteristic of these schemes is that it reduces the original equation into the system of nonlinear algebraic equations and thus greatly simplifying the problem. This system is then solved to find the unknown coefficients. The working of proposed schemes is illustrated on several test examples and the obtained numerical results confirm the desired accuracy and efficiency of the schemes. Further, we investigated the convergence analysis and also established the error bounds for the proposed schemes. Finally, it is found that both the schemes are easy to implement, but scheme-I produces better numerical results and also takes less CPU time in comparison to scheme-II. • Approximate solutions of nonlinear weakly singular fractional integro-differential equations in 1D and 2D are discussed. • Two new schemes based on the shifted Legendre polynomials (SLP) and interpolating basis functions (IBF) are constructed. • Error bounds for function approximations are established for presented schemes. • Convergence analysis is rigorously discussed for both the schemes. • The accuracy and efficiency of the presented schemes are illustrated by performing numerical experiments on several test functions.

Numerical Scheme with Convergence Analysis and Error Estimate for Variable Order Weakly Singular Integro-Differential Equation

This paper represents a new application of Legendre wavelet and interpolating scaling function to discuss the approximate solution of variable order integro-differential equation having weakly singular kernel. So far, this technique has been used to solve variable order integro differential equation. In this paper, it is extended to solve variable order integro differential equation with weakly singular kernel. For this purpose, we derive the operational matrices of Legendre wavelets and interpolating scaling function. The resulting operational matrices along with the collocation method transform the original problem into a system of algebraic equation. By solving this system, the approximate solution is obtained. The convergence and error estimate of the presented method have been rigorously investigated. We also discuss the numerical stability of the method. The numerical result of some inclusive examples has been provided through a table and graph for both basis functions that support the robustness and desired precision of the method.