Solving fractional integral equations by the Haar wavelet method

Haar wavelets for the solution of fractional integral equations are applied. Fractional Volterra and Fredholm integral equations are considered. The proposed method also is used for analysing fractional harmonic vibrations. The efficiency of the method is demonstrated by three numerical examples.     

Qualitative properties of solutions of integral equations

In this paper we study a linear integral equation in which the kernel fails to satisfy standard conditions yielding qualitative properties of solutions. Thus, we begin by following the standard idea of differentiation to obtain . The investigation frequently depends on x^′(t)+C(t,t)x(t)=0 being uniformly asymptotically stable. When that property fails to hold, the investigator must turn to ad hoc methods. We show that there is a way out of this dilemma. We note that if C(t,t) is bounded, then for k>0 the equation resulting from x^′+kx will have a uniformly asymptotically stable ODE part and the remainder can often be shown to be a harmless perturbation. The study is also continued to the pair x^″+kx^′.     

The Kontorovich-Lebedev integral transformation with a Hankel function kernel in a space of generalized functions of doubly exponential descent

A version of the Kontorovich-Lebedev transformation with the Hankel function of second kind in the kernel is investigated in a space of distributions of doubly exponential descent. The inversion theorem is rigorously established making use in some steps of the proof of a relation of this transform with the Laplace one. Finally, the theory developed is illustrated in solving certain type of partial differential equations.     

A new type of integral equations related to the co-area formula (Reduction of dimension in multi-dimensional integral equations)

Some classes of multi-dimensional integral equations of the second kind are introduced which admit an equivalent reduction to integral equations for functions depending on a smaller number of independent variables. This reduction is performed by factorization of the integral operator based on the co-area formulas.     

Solving boundary value problems,integral, and integro-differential equations using Gegenbauer integration matrices

We introduce a hybrid Gegenbauer (ultraspherical) integration method (HGIM) for solving boundary value problems (BVPs), integral and integro-differential equations. The proposed approach recasts the original problems into their integral formulations, which are then discretized into linear systems of algebraic equations using Gegenbauer integration matrices (GIMs). The resulting linear systems are well-conditioned and can be easily solved using standard linear system solvers. A study on the error bounds of the proposed method is presented, and the spectral convergence is proven for two-point BVPs (TPBVPs). Comparisons with other competitive methods in the recent literature are included. The proposed method results in an efficient algorithm, and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems. The proposed method can be applied on a broad range of mathematical problems while producing highly accurate results. The developed numerical scheme provides a viable alternative to other solution methods when high-order approximations are required using only a relatively small number of solution nodes.     

Smoothed integral equations

For a linear integral equation there is a resolvent equation and a variation of parameters formula . It is assumed that B is a perturbed convex function and that a(t) may be badly behaved in several ways. When the first two equations are treated separately by means of a Liapunov functional, restrictive conditions are required separately on a(t) and B(t,s). Here, we treat them as a single equation where S is an integral combination of a(t) and B(t,s). There are two distinct advantages. First, possibly bad behavior of a(t) is smoothed. Next, properties of S needed in the Liapunov functional can be obtained from an array of properties of a(t) and B(t,s) yielding considerable flexibility not seen in standard treatment. The results are used to treat nonlinear perturbation problems. Moreover, the function is shown to converge pointwise and in L²[0,∞) to x(t).     

Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations

Some new weakly singular integral inequalities of Gronwall-Bellman type are established, which generalized some known weakly singular inequalities and can be used in the analysis of various problems in the theory of certain classes of differential equations, integral equations and evolution equations. Some applications to fractional differential and integral equations are also indicated.     

An integral recursive inequality and applications

An integral recursive inequality for two functions is obtained. It is used to describe the equality cases in the related inequalities. The applications involve some bi-Hermitian forms, integral transformations, and confluent hypergeometric functions.      

Imbedding methods for integral equations with applications

During the last decade or two, significant progress has been made in the development of imbedding methods for the analytical and computational treatment of integral equations. These methods are now well known in radiative transfer, neutron transport, optimal filtering, and other fields. In this review paper, we describe the current status of imbedding methods for integral equations. The paper emphasizes new analytical and computational developments in control and filtering, multiple scattering, inverse problems of wave propagation, and solid and fluid mechanics. Efficient computer programs for the determination of complex eigenvalues of integral operators, analytical investigations of stability for significant underlying Riccati integrodifferential equations, and comparisons against other methods are described.     

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

In this paper, we study the numerical solution to time‐fractional partial differential equations with variable coefficients that involve temporal Caputo derivative. A spectral method based on Gegenbauer polynomials is taken for approximating the solution of the given time‐fractional partial differential equation in time and a collocation method in space. The suggested method reduces this type of equation to the solution of a linear algebraic system. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations

In this paper, existence and attractiveness of solutions for quadratic Urysohn fractional integral equations on an unbounded interval are obtained by virtue of Tichonov fixed point theorem and suitable conjunction of the well known measure ω₀(X) and the spaces C(R⁺). Further, three certain solutions sets X_L,γ, X_1,α and X_{1,(1−(α+v))}, which tending to zero at an appropriate rate t^−ν (ν > 0), ν = γ (or α or 1 − (α + v)) as t → ∞, are introduced and stability of solutions for quadratic Urysohn fractional integral equations are obtained based on these solutions sets respectively by applying Schauder fixed point theorem via some easy checked conditions. An example is given to illustrate the results.     

Operational solution of fractional differential equations

The main goal of this paper is to solve fractional differential equations by means of an operational calculus. Our calculus is based on a modified shift operator which acts on an abstract space of formal Laurent series. We adopt Weyl’s definition of derivatives of fractional order.     

Hybrid function method for solving Fredholm and Volterra integral equations of the second kind

Numerical solutions of Fredholm and Volterra integral equations of the second kind via hybrid functions, are proposed in this paper. Based upon some useful properties of hybrid functions, integration of the cross product, a special product matrix and a related coefficient matrix   with optimal order, are applied to solve these integral equations. The main characteristic of this technique is to convert an integral equation into an algebraic; hence, the solution procedures are either reduced or simplified accordingly. The advantages of hybrid functions are that the values of n$n$ and m$m$ are adjustable as well as being able to yield more accurate numerical solutions than the piecewise constant orthogonal function, for the solutions of integral equations. We propose that the available optimal values of n$n$ and m$m$ can minimize the relative errors of the numerical solutions. The high accuracy and the wide applicability of the hybrid function approach will be demonstrated with numerical examples. The hybrid function method is superior to other piecewise constant orthogonal functions [W.F. Blyth, R.L. May, P. Widyaningsih, Volterra integral equations solved in Fredholm form using Walsh functions, Anziam J. 45 (E) (2004) C269–C282; M.H. Reihani, Z. Abadi, Rationalized Haar functions method for solving Fredholm and Volterra integral equations, J. Comp. Appl. Math. 200 (2007) 12–20] for these problems.     

An alternative method to obtain integral kernels for Lamé integral equations

In the present paper we introduce an alternative approach to obtain kernels for Lamé integral equations. The introduced procedure, based solely on simple algebraic manipulations, furnishes the well known kernels of hypergeometric form in an almost trivial manner without the use of transformations. More important, it provides a set of novel nuclei for Lamé integral equations in the form of Heun functions.     

Simplified reproducing kernel method for fractional differential equations with delay

This paper is devoted to the numerical scheme for the delay initial value problems of a fractional order. The main idea of this method is to establish a novel reproducing kernel space that satisfies the initial conditions. Based on the properties of the new reproducing kernel space, the simplified reproducing kernel method (SRKM for short) is applied to obtain accurate approximation. The Schmidt orthogoralization process which requires a large number of calculation is less likely to be employed. Numerical experiments are provided to illustrate the performance of the method.     

A unification theory of Krasnoselskii for differential equations

The theory of differential equations is very broad and contains many seemingly unrelated types of problems with markedly different methods of solution. It is very difficult to discern any unity in the theory. Yet, sixty years ago one of the foremost investigators, Krasnoselskii, suggested the possibility of finding unity. He claimed that the inversion of a perturbed differential operator yields the sum of a contraction and compact map. Accordingly, he proved a general fixed point theorem to cover this situation. In this paper we begin a long study with a view to putting his idea to the test. We begin with fractional differential equations of Caputo type, continue to neutral functional differential equations, and conclude with a study of an old problem of Volterra which continues to describe many important real-world problems. For these problems there is the perfect unity predicted by Krasnoselskii. It is an invitation to continue the study by examining other important real-world problems.     

Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives

In this paper, by introducing the fractional derivative in the sense of Caputo, the Adomian decomposition method is directly extended to study the coupled Burgers equations with time- and space-fractional derivatives. As a result, the realistic numerical solutions are obtained in a form of rapidly convergent series with easily computable components. The figures show the effectiveness and good accuracy of the proposed method.     

Henry-Gronwall type retarded integral inequalities and their applications to fractional differential equations with delay

This paper presents retarded integral inequalities of Henry-Gronwall type. Applying these inequalities, we study certain properties of solutions to fractional differential equations with delay.     

Calderon reproducing formula associated with the Gegenbauer operator on the half-line

In this paper, we introduce the generalized shift operator generated by the Gegenbauer differential operator , and define a generalized convolution ⊗ on the half-line corresponding to the Gegenbauer differential operator. We investigate the Calderon reproducing formula associated with the convolution ⊗ involving finite Borel measures, leading to results on the Lp-norm and pointwise approximation for functions on the half-line.     

Generalized integral operators and Schwartz kernel type theorems

In analogy to the classical Schwartz kernel theorem, we show that a large class of linear mappings admits integral kernels in the framework of Colombeau generalized functions. To do this, we introduce new spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that, in some sense, Schwartz' result is contained in our main theorem.