SOLUTION OF VOLTERRA-TYPE INTEGRO-DIFFERENTIAL EQUATIONS WITH A GENERALIZED LAURICELLA CONFLUENT HYPERGEOMETRIC FUNCTION IN THE KERNELS

The object of this paper is to solve a fractional integro-differential equation involving a generalized Lauricella confluent hypergeometric function in several complex variables and the free term contains a continuous function f(τ). The method is based on certain properties of fractional calculus and the classical Laplace transform. A Cauchy-type problem involving the Caputo fractional derivatives and a generalized Volterra integral equation are also considered. Several special cases are mentioned. A number of results given recently by various authors follow as particular cases of formulas established here.


Introduction and preliminaries
The first-order integro-differential equation of Volterra type [7,9]  describes the unsaturated behavior of the free electron laser (FEL).Here τ is a dimensionless time variable, g 0 is a positive constant called the small signal gain, and the detuning parameter is the constant ν.The function a(τ) is the complex-field amplitude which is assumed to be dimensionless satisfying the initial condition a(0) = 1.The exact closed form solution of (1.1) valid in the whole range of practical interest and suitable for numerical calculations was given by Dattoli et al. [8].
Fractional calculus has gained importance during the last three decades or so due to its various applications in the solution of fractional differential and fractional integral equations arising in various problems of physics engineering and applied sciences, such as diffusion in porous media, fractal geometry, kinematics in viscoelastic media, propagation of seismic waves, anomalous diffusion, and so forth.In this connection, one can refer to the works mentioned in [6,12,16,18,20,21,25].
The Riemann-Liouville operator of fractional integration of order ν is defined by 1156 Solution of Volterra-type equations The Riemann-Liouville fractional derivative of order ν is defined in the form [19,20,21,25] (1.3) By the application of the convolution theorem of the Laplace transform [11, page 131], we find from (1.2) that where H(s) is the Laplace transform of h(t) defined by L h(t) : s = ∞ 0 h(t)e −st dt =: H(s), Re(s) > 0, (1.5) which may be written symbolically as follows: provided that the function h(t) is continuous for t ≥ 0 and of exponential order as t → ∞.Boyadjiev et al. [5, (7), page 4] studied the following nonhomogeneous form of fractional integro-differential equation of Volterra type: where β,λ ∈ C and ν ∈ R.
Al-Shammery et al. [3, (14), page 82] considered a generalization of (1.4) in the form Al-Shammery et al. [2, (14), page 503] further studied another generalization of (1.7) in the form where 0 Saxena and Kalla [26] derived the solution of a further generalizaion of (1.9) in the form where R. K. Saxena and S. L. Kalla 1157 Recently Kilbas et al. [13] systematically studied a generalization of (1.10) in the following form: where a ≤ x ≤ b; λ,µ,ρ,γ ∈ C; ω ∈ R, Re(α) > 0, Re(µ) > 0, and f is assumed to be Lebesgue integrable over the interval (a,b) and the function where Re(ρ) > 0; (γ A comprehensive account of the various generalizations of FEL equations is recently given in a survey paper by Boyadjiev and Kalla [4].A description of various special functions appearing in this paper is available in [1,10,11,31].
A detailed account of various operators of fractional integration and their applications can be found in a recent survey paper of Srivastava and Saxena [32].An interesting account of convolution integral equations has been given by Srivastava and Buschman [28].
The main object of the present paper is to introduce a further generalization of (1.11) in an interesting and unified form in which the generalized Mittag-Leffler function, the involved kernel, is replaced by a generalized Lauricella confluent hypergeometric function in several complex variables and the free term contains a continuous function f (τ).The method followed here in finding the solution of Cauchy-type problem (3.1) and (3.2) and other Cauchy problems is based upon certain properties of fractional calculus and the classical Laplace transform.The solutions derived are in closed forms and are suitable for numerical computation.

Integral representations of generalized Lauricella confluent hypergeometric function of several complex variables
The following result is quite interesting and useful as it gives the inverse Laplace transforms of the product of binomial functions: is defined for complex λ,γ j ,ρ j ,ω j ,z j (Re(ρ j ) > 0) j = 1,...,n, in terms of a multiple series in the following form: (2.5) Here the contours Ω j 's are given by Ω j = Ω isj ∞ (i = (−1) 1/2 , Re(s j ) = ν j ) starting at the point ν j − i∞ and terminating at the point ν j + i∞; ν j ∈ ( j = 1,...,n).All the poles of the gamma functions appearing in the integrand of (2.5) are assumed to be simple.
When n = 1, (2.1) reduces to the generalized Mittag-Leffler function due to Prabhakar [22] defined by (1.12), consequently the results obtained in this paper will generalize the work reported earlier by Kilbas et al. [13].
Remark 2.2.A detailed and comprehensive account of the multiple Gaussian hypergeometric functions is available from Srivastava and Karlsson [31].
Solving (3.6) under the initial conditions (3.2), we find that where it is tacitly assumed that By the application of the formula (2.1) once again, it is found from (3.7) that (3.9) The above expression can be expressed in the form (3.3).This completes the proof of Theorem 3.1.
In order to establish the uniqueness of the solution (3.3), we set ζ → τ − ζ in (3.1) and operate upon both sides by D −α τ (Re(α) > 0).Next, if we apply the Dirichlet formula [25] and make use of the Eulerian integral for the Beta function, we arrive at the following equation: (3.10)Since (3.10) is a Volterra integral equation with a continuous kernel, it does admit a unique continuous solution (see [15]).
Remark 3.2.The solution of the Cauchy-type problem (3.1) and (3.2) can also be developed by the method of successive approximations.In this connection, see [13,23].

If we set
then by virtue of the identity (2.10), we arrive at the following result recently obtained by Srivastava and Saxena [33].

Corollary 4.2. Under the various relevant hypotheses of Theorem 3.1, a unique continuous solution of the Cauchy-type problem involving the Volterra-type integro-differential equation
,Re(γ),Re(ω),Re(µ)} > 0, together with the initial conditions (3.2), is given by , together with the initial conditions (3.2), is given by where Remark 4.4.In its further special case, when where ) and Λ k 's are given in (3.4).

A Cauchy-type problem involving the Caputo fractional derivatives
In connection with certain investigations, especially in the theory of viscoelasticity and hereditary solid mechanics, Caputo introduced the following definition for the fractional derivative of order α > 0 of a casual function f (t) (i.e., f (t) = 0, for t < 0), which arose in several important earlier works (see, for details [21, page 78]): where h (m) (t) denotes the usual (ordinary) derivative of h(t) of order m (m ∈ N 0 ).It readily follows from the definitions (1.5) and (5.1) that which is preferred for initial-value problems of physical sciences than (2.17), H(s) is given by (1.5).

Solution of the generalized Volterra integral equation
In this section, we present a generalization of the Volterra integral equation quite recently given by Srivastava and Saxena [33].
1168 Solution of Volterra-type equations When ρ 1 = ••• = ρ n = 1, then using the relation (2.12), it immediately yields the following result recently given by Srivastava and Saxena [33], which itself is a generalization of Srivastava's result [27].Corollary 6.2.The Volterra-type integral equation has its solution given explicitly by

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning − σ)exp(iνσ)dσ(1.1) and Φ(a,b;z) is the Kummer confluent hypergeometric function defined in [10, (1), page 248].

Corollary 4 . 1 .
Under the various relevant hypotheses of Theorem 3.1, a unique continuous solution of the Cauchy-type problem involving the Volterra-type integro-differential equation

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation