- Research
- Open Access

# Operational method of solution of linear non-integer ordinary and partial differential equations

- K. V. Zhukovsky
^{1}Email author

**Received:**10 November 2015**Accepted:**15 January 2016**Published:**9 February 2016

## Abstract

We propose operational method with recourse to generalized forms of orthogonal polynomials for solution of a variety of differential equations of mathematical physics. Operational definitions of generalized families of orthogonal polynomials are used in this context. Integral transforms and the operational exponent together with some special functions are also employed in the solutions. The examples of solution of physical problems, related to such problems as the heat propagation in various models, evolutional processes, Black–Scholes-like equations etc. are demonstrated by the operational technique.

## Keywords

- Inverse operator
- Derivative
- Differential equation
- Special functions
- Hermite and Laguerre polynomials

## PACS Nos.

- 02.30 Gp
- Hq
- Jr
- Mv
- Nw
- Tb
- Uu
- Vv
- Zz
- 41.85.Ja
- 03.65.Db
- 05.60.Cd

## Background

Differential equations, besides playing important role in pure mathematics, constitute fundamental part of mathematical description of physical processes. Thus, obtaining the solutions for differential equations is of paramount importance. Few types of differential equations allow explicit and straightforward analytical solutions. The development of computer methods and proper technical means in twenty-first century facilitated equations solving. There are numerous numerical methods for solving differential equations (see, for example, Von Rosenberg 1969; Smith 1985; Ghia et al. 1982; Ames 2014; Johnson 2012; Carnahan et al. 1969). However, understanding of the obtained solutions and of the interplay of various parameters in them can be best done in analytical form. Thus, despite the revolutionary breakthrough in numerical calculus, analytical studies remain requested. Exact and approximate solutions are searched, while the first are certainly preferred. Recently some fractional type ordinary and partial differential equations involving non-integer derivatives were explored in Demiray et al. (2015). Exact analytical solutions for such equations were obtained as sums of a vector-type functional (Akinlar and Kurulay 2013). Some partial differential–algebraic equations were also solved by the power series method (Filobello-Nino et al. 2015; Benhammouda and Vazquez-Leal 2014). The solutions were obtained in the form of converging series. Fokker–Planck type equations were recently solved by differential transforms in Hesam et al. (2012); the solutions obtained in form of the form of rapidly convergent series. Semi-analytical techniques for the solution of differential–algebraic equations were developed (Soltanian et al. 2013) and applied for description of an incompressible viscous fluid flow. Approximate solutions for some nonlinear delay differential equations were obtained (Caruntu and Bota 2014) and applied to a biologic model. A modified method of simplest equation was proposed in Vitanov et al. (2015) to find exact analytical solutions of nonlinear partial differential equations. In many cases, these solutions are best formulated in terms of special functions and orthogonal polynomials when used for relevant models of physical processes. Hyperbolic, elliptic Weierstrass and Jacobi type, generalized Airy and Bessel type functions are used (Vitanov et al. 2015; Dattoli et al. 2008, 2009; Appèl and Kampé de Fériet 1926; Dattoli 2000; Dattoli et al. 2005; Zhukovsky 2014, 2015a, b, c, d); expansion in series of Hermite and Laguerre polynomials (Appèl and Kampé de Fériet 1926) are employed. These polynomials possess generalized forms with many variables and indices (Dattoli 2000; Dattoli et al. 2005). In this framework the operational definitions for the polynomials are useful (Erdélyi et al. 1953). The above mentioned recent developments in analytical and semi-analytical equations solutions (Demiray et al. 2015; Akinlar and Kurulay 2013; Filobello-Nino et al. 2015; Benhammouda and Vazquez-Leal 2014; Hesam et al. 2012; Soltanian et al. 2013; Caruntu and Bota 2014; Vitanov et al. 2015) are indeed capable of reducing the size of computational work.

In what follows we shall demonstrate the abilities of the operational approach for solution of differential equations. With the help of this general method we will obtain exact analytical solutions for a broad class of differential equations, including those with non-integer derivatives, evolution type equations, generalized forms of heat, mass transfer and Black–Scholes type equations, involving also the Laguerre derivative operator. Recently, this method was applied for solution of some differential equations in Zhukovsky (2014, 2015a) and Dattoli et al. (2007). These equations cover a broad range of physical problems: from propagation and radiation of accelerated charges to heat and mass transfer (see, for example, Haimo and Markett 1992a, b; Zhukovsky 2016). Operational exponent, employed for solution, finds its application even for description of such fundamentals of nature as quarks and neutrinos (Dattoli and Zhukovsky 2007a, b, 2008). We will not include error analysis in our work since the proposed operational method produces analytical solutions, which satisfy the equations exactly.

When it comes to a numerical analysis, there are also practical and theoretical reasons for examining the process of inverting differential operators. Indeed, the inverse or integral form of a differential equation displays explicitly the input–output relationship of the system. Moreover, integral operators are computationally and theoretically less troublesome than differential operators; for example, differentiation emphasizes data errors, whereas integration averages them. Thus, it may be advantageous to apply computational procedures to differential systems, based on the inverse or integral description of the system.

The evident concept of an inverse function is a function that undoes another function: if an input *x* into the function *ƒ* produces an output *y*, then putting *y* into the inverse function *g* produces the output *x*, and vice versa. i.e.,\( f(x) = y \) and \( g(y) = x \) or \( g(f(x)) = x \). If a function ƒ has an inverse *ƒ*
^{−1}, it is invertible and the inverse function is then uniquely determined by ƒ. We can develop similar approach with regard to differential operators. In what follows we will further develop this technique and explore its relation with extended forms of orthogonal polynomials, producing useful relations for solution of a variety of differential equations, by means of inverse derivative. The relevant physical problems will be considered.

*ƒ*(

*x*) is another function

*F*(

*x*): \( D^{ - 1} f(x) = F(x) \), whose derivative is \( F^{\prime}(x) = f(x) \). Naturally, we expect anti-derivative or inverse derivative \( D^{ - 1} \) as the inverse operation of differentiation to be an integral operator. The generalized form of the inverse derivative of

*ƒ*(

*x*) with respect to

*x*evidently is \( \int {f(x) = F(x) + C} \), where

*C*—the constant of integration. The action of the inverse derivative operator of the

*n*-th order

*D*,

*D*

^{2}, …,

*D*

^{n}the inverse differential operator \( 1/\psi (D) \) or \( (\psi (D))^{ - 1} \) is defined, such that

*a*and

*b*are constants,

*ƒ*(

*x*) and

*g*(

*x*) are some functions of

*x*. Let us consider an elementary example of the following simple equation:

*αx*) results in \( \psi (D)e^{\alpha x} = (c_{n} D^{n} + \cdots + c_{1} D + c_{0} )e^{\alpha x} = \psi (\alpha )e^{\alpha x} \); applying inverse operator \( (\hat{\psi }(D))^{ - 1} \) to both sides, we obtain: \( e^{ax} = \psi (\alpha )\frac{1}{\psi (D)}e^{\alpha x} \) or \( \frac{{e^{ax} }}{\psi (\alpha )} = \frac{1}{\psi (D)}e^{\alpha x} \) and we conclude that Eq. (8) possesses the following particular integral:

### Inverse differential and exponential operators for solution of some non-integer ordinary differential equations

*α*,

*β*—constants. In order to find the particular integral

*α*= 0

*y*and

*α*are the parameters. It eventually yields the following solution for Eq. (12):

*ν*, is a result of consequent action of operators of heat propagation \( \hat{S} \) and operator of translation \( \hat{\varTheta } \) on the function \( f(x) \):

So far, we have demonstrated on simple examples how the usage of inverse derivative together with operational formalism and, in particular, with exponential operator technique, provide elegant and easy way to find solutions in some classes of differential equations. In what follows we will apply the concept of inverse differential operator to find solutions of more sophisticated problems, expressed by differential equations.

### Operational approach and orthogonal polynomials for solution of some non-integer ordinary differential equations

Various polynomial families, such as Hermite, Laguerre, Legendre, Shaffer and hybrid polynomials can be reviewed in the context of umbral calculus as members of a more general family of Appèl polynomials, which they belong to. Such consideration is possible in the framework operational approach, where inverse derivative plays important role as an instrument for the study of relevant polynomial families, their features and properties.

*α*,

*β*= 0, obviously follow from (54).

*f*in the r.h.s. of (12) and the values of

*ν*and

*α*and, we can still disentangle two integrals in (21) by involving Hermite polynomials of two variables (40) as follows:

### Operational solution of some partial differential equations

The method of the inverse differential operators has multiple applications for solving mathematical problem, describing wide range of physical processes, such as the heat transfer, the diffusion, wave propagation etc. Some of the examples of solution of the heat equation, of the diffusion equation and of their modified forms, the Laguerre heat equation and others, by the inverse derivative method were considered in Dattoli et al. (2006, 2007) and Zhukovsky and Dattoli (2011). In what follows we will explore more complicated, generalized forms of the aforementioned equations, as well as some second order over the time variable partial differential equations will be touched on.

It is worth mentioning that, despite the relation (11) seems trivial to all appearance, it is very useful for solution of a broad family of differential equations by operational method. Indeed, for the differential equation \( \psi (D_{x} + \alpha )F(x,t) = f(x,t) \) we can rewrite (11) in the following form: \( e^{\alpha x} F(x,t) = \psi^{ - 1} (D_{x} )e^{\alpha x} f(x,t) \) and, for example, for the evolutional type equations, where \( f(x,t) = \partial_{t} F(x,t) \), we obtain \( \psi (D_{x} )e^{\alpha x} F(x,t) = \partial_{t} e^{\alpha x} F(x,t) \). By denoting \( e^{\alpha x} F(x,t) = G(x,t) \) we have the equation \( \psi (D_{x} )G(x,t) = \partial_{t} G(x,t) \) with \( \psi (D_{x} ) \) and with the initial condition \( g(x) = G(x,0) = e^{\alpha x} F(x,0) = e^{\alpha x} f(x) \). Thus, in order to obtain the desired solution \( F(x,t) = e^{ - \alpha x} G(x,t) \) of the equation \( \psi (D_{x} + \alpha )F(x,t) = f(x,t) \) with the initial condition \( F(x,0) = f(x) \), we end up with the necessity to solve the equation with \( \psi (D_{x} ) \) for the function \( G(x,t) \) with the initial condition \( g(x) = e^{\alpha x} f(x) \). Note that the above discussed method is applicable not only to the evolutional type equations with \( \partial_{t} \) in the r.h.s., but also to other operators \( \hat{D}\left( t \right) \), acting over the time variable. Indeed, if \( G\left( {x,t} \right) \) is the solution of \( \psi (D_{x} )G(x,t) = \hat{D}\left( t \right)G(x,t) \) with \( g(x) = G(x,0) = e^{\alpha x} F(x,0) = e^{\alpha x} f(x) \), then, following the above scheme, it is easy to demonstrate that \( F(x,t) = e^{ - \alpha x} G(x,t) \) is the solution of the equation \( \psi (D_{x} + \alpha )F(x,t) = \hat{D}\left( t \right)F(x,t) \) with \( F(x,0) = f(x) \). Evidently, in the case of the second order differential operator \( \hat{D}\left( t \right) \) the second boundary or initial condition has to be chosen for the differential equation for \( F\left( {x,t} \right) \) and, accordingly, for \( G\left( {x,t} \right) \). In what follows, we shall apply the above-discussed method to several examples of equations, common in physics and not only.

#### Black–Scholes type equations

*ρ*,

*λ*andи

*μ*are the constant coefficients and \( f\left( x \right) = F\left( {x,0} \right) \) is the initial condition function. The apparently complicated Eq. (76) reduces to the following form:

*ρ*,

*λ*and

*μ*are the constant coefficients and \( g\left( x \right) = A\left( {x,t = 0} \right) \) is the initial condition function. The Eq. (81) generalizes and unifies equations of Laguerre diffusion of matter and of heat, considered in Dattoli et al. (2005, 2007). This equation also can be solved by the operational method developed above. Indeed, by distinguishing the perfect square of the Laguerre derivative \( {}_{L}D{}_{x} = \partial_{x} x \partial_{x} \) in (81), the solution evidently reads in the form of the exponential \(A\left( {x,t} \right) = \exp \left\{ {\rho t\left( {(_{L} D_{x} + \lambda /2)^{2} - \varepsilon } \right)} \right\}g(x) \), where \( \varepsilon = \mu + \left( {\lambda /2} \right)^{2} \). Now we apply the operational identity (16) to \( \exp (a_{L} D_{x} ) \) to obtain the following solution for \( A(x,t) \):

#### Heat diffusion type equations

*x*: \( f\left( x \right) = x^{k} \). Then \( g\left( x \right) = x^{k} e^{\delta x} \) and we have in fact Eq. (91) for \( G \): \( \partial_{t} G = (\partial_{x}^{2} + \beta x)G \). Upon the action of the \( \hat{\bar{S}} \) operator on it and due to the operational rule (52) we obtain \( \hat{S}g\left( x \right) = \exp \left( {\delta \left( {x + \delta a} \right)} \right)H_{k} \left( {x + 2\delta a,a} \right) = g\left( {x,t} \right) \). The consequent action of the translation operator \( \hat{\varTheta } \) yields the shift along the

*x*argument and, thus, the solution of Eq. (91) for \( G \), taken \( G\left( {x,0} \right) = g\left( x \right) \), has the following form:

*x*and

*y*: \( f\left( {x,y} \right) = x^{m} y^{n} \). Then, according to the operational definition of the Hermite polynomials of four variables and two indices \( H_{m,n} \left( {\left. {x,t\alpha ,y,t\gamma } \right|\beta } \right) \) (see, for example, Erdélyi et al. 1953; Dattoli et al. 2006, 2007) we obtain

### Operational solution of some second order of time partial differential equations

### Hyperbolic heat equation solution

*C*in media (\( \tau = k_{T} /C^{2} \)); \( \sqrt {k_{T} /\tau } = C \) represents a velocity like quantity, associated with the speed of the heat wave in the medium, which characterizes the thermal wave propagation the same way as the diffusion behaviour is characterized by the diffusivity. Equation (121) is the simplest model of the second sound phenomenon observed first in liquid Helium (Peshkov 1944) and then also in solid crystals (Ackerman and Overton 1969). To solve it we have to compute the result of the action of the operator \( \hat{S} \): \( e^{{ - 4\alpha \xi \partial_{x}^{2} }} f\left( x \right) \). The fading at infinite time solution for the initial function \( f\left( x \right) \) follows directly from (109):

*n*and

*γ*, for the example for \( n = - \gamma = 1 \), we obtain

Non-Fourier diffusive, wave-like heat propagation in Cattaneo’s model (120), (121) had some success in the description of second sound. However, it did not agree with the experimental observations and it was superseded by other, more adequate heat propagation models, which included additional terms in the hyperbolic equation to describe the whole complex of phenomena. We will obtain analytical solutions for them in forthcoming publications.

## Results and conclusions

We advocated operational method for solution of linear differential equations. We use inverse differential operators; it allows direct and straightforward computation of solutions in the framework of operational calculus. The obtained solutions contain consequent action of operators of heat conduction and shift, involving common and Laguerre derivative with exponential and power factors. Their action can be expressed via Gauss type integrals and shift of arguments. Using operational definitions of Hermite and Laguerre orthogonal polynomial families, we executed direct operational transforms over them, consisting in shift and factorization. Combined where necessary with integral transforms, it yielded solutions of relatively complicated linear differential equations of several types.

In particular, we have obtained explicit exact solutions for some ordinary differential equations of non-integer dimension, involving shifted derivatives. The particular solution of equation \( \psi^{ - 1} (D) = \left( {\beta^{2} - (D + \alpha )^{2} } \right)^{ - \nu } f(x) \) for any Real ν is given by the integral of the weighted consequent action of operators of heat propagation \( \hat{S} \) and translation \( \hat{\varTheta } \) on the function \( f(x) \). We wrote it as a convolution transform \( \phi \left( {x,\tau } \right) = G\left( {x,\tau } \right) \cdot f\left( \eta \right) \) or \( \phi = \int\nolimits_{ - \infty }^{\infty } {G\left( {x - \eta } \right)f\left( \eta \right)d\eta } \) with the kernel, equal to the Gauss frequency function. The examples of Gaussian \( f(x) = e^{{ - x^{2} }} \) and monomial functions \( f(x) = x^{k} \) were demonstrated by explicit solutions in terms of integrals and series of Hermite polynomials. Operational solutions for equations, involving Laguerre derivatives: \( (x\partial^{2}_{x} + \left( {\alpha + 1} \right)\partial_{x} )^{\nu } F(x) = f(x) \) were obtained. Examples of the exponential \( f(x) = \exp ( - \gamma x) \), of the monomial and of Bessel–Wright functions were demonstrated. By using operational technique we immediately write their explicit solutions; they involve integrals and generalized Laguerre polynomials.

We obtained solutions for several types of partial differential equations. In particular, extended Black–Scholes equation was solved. Moreover, generalized form of Black–Scholes type equation with Laguerre derivative \( {}_{L}D_{x} = \partial_{x} x \partial_{x} \) was solved operationally. The example of a monomial initial function yields the explicit solution with series of gamma function and hypergeometric function. For initial Bessel–Wright function the solution of the Black–Scholes equation with Laguerre derivative is given by integrals of Bessel–Tricomi function.

Extended forms of heat diffusion equation were solved. Their operational solution readily yields explicit forms upon consequent action of operators of shift and heat diffusion on the initial function. Examples of initial functions \( g\left( x \right) = x^{k} e^{\delta x} \) and \( f\left( {x,y} \right) = x^{m} y^{n} \) produce Hermite polynomials and their generalized forms with four variables and two indices: \( H_{m,n} \left( {\left. {x,t\alpha ,y,t\gamma } \right|\beta } \right) \). Two-dimensional heat diffusion type equation with the linear terms was solved. Its operational solution consists in the action of the generalized two-dimensional analogue of heat diffusion operator and respective coordinate shifts with a phase factor.

Operational solutions of a number of hyperbolic equations with regular and Laguerre coordinate derivatives were obtained. For the second order of time hyperbolic equations with Laguerre derivatives of the 1st and 2nd order we obtained explicit solutions in terms of elementary functions, Gegenbauer polynomials and gamma function for the initial monomial \( f\left( x \right) = x^{n} \) and for the exponential \( f\left( x \right) = x^{m} e^{ - \alpha x} \). Hyperbolic heat equation was thoroughly explored with the help of operational method. Explicit solution for \( f\left( x \right) = x^{m} e^{ - \alpha x} \) was obtained in elementary functions. The role of various equation terms in the behaviour of the solution was elucidated. Maintaining the heat conductivity unchanged, we underlined the role of the second time derivative. Fading of the wave propagation happens earlier, if we reduce the effect of the second time derivative in the equation, choosing its coefficient small with respect to others: \( k_{T} = 1,\;\tau = 0.1 \). In this case the diffusive character of the heat conduction prevails over the wave-like propagation process. On the contrary, high value of the second time derivative term in (120): \( k_{T} = 1,\,\tau = 10 \), underlines the wave-like propagation with very little fading as follows from the comparison of Figs. 1, 2 and 3.

Thus, operational approach allowed for easy and straightforward solution of differential equations and relevant physical problems, such as modified Fourier heat diffusion in three dimensions, Cattaneo heat propagation, Laguerre type diffusion, evolution of a system, obeying Black–Scholes type equations, common in financial studies. Operational method has obvious advantages, respectively to other methods: it is universal, applies to ordinary and partial linear DE and non-integer DE, the solutions are obtained readily; they are light computationally and have transparent meaning. The effect of each term in the initial equation on the solution is distinguished. The validity of the obtained solutions was verified by direct substitution in the solved equations. The solutions in form of integrals contain consequent action of operators of heat conduction and shift with exponential and power factors. The considered examples of solutions of the hyperbolic and Fourier heat equations with common and Laguerre derivatives for given initial functions contain integrals and series of Hermite polynomials; explicit solutions, such as (96), (97), (105), (113), (119), (126) do not possess critical or singular coordinate points. For the second order of time differential Eqs. (106) with ordinary and Laguerre derivatives we obtained strictly bounded solutions. Rigorous investigation of stability of all of the obtained solutions for different initial functions, including fractional differential equations and employing Lyapunov methods, will constitute a stand-alone study in a forthcoming dedicated publication.

In conclusion we would like to note, that our results, being exact, can represent a benchmark for numerical solutions. These latter can, perhaps, cover more extensions, but should reduce to our results in the limiting cases. Application of our study for more complicated equations, describing non-Fourier heat propagation by ballistic heat transfer and other equations, modelling physical processes, will be also made in forthcoming publications.

## Declarations

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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