On Some Generalizations of the Properties of the Multidimensional Generalized Erdélyi–Kober Operators and Their Applications

Abstract

In this paper we investigate the composition of a multidimensional generalized Erdélyi–Kober operator with differential operators of high order. In particular, with powers of the differential Bessel operator. Applications of proved properties to solving the Cauchy problem for a multidimensional polycaloric equation with a Bessel operator are shown. An explicit formula for solving the formulated problem is constructed. In the appendix we briefly describe a general context for transmutations and integral transforms used in this paper. Such a general context is formed by integral transforms composition method (ITCM).

Appendix: Integral Transform Composition Method (ITCM) in Transmutation Theory: How It Works

In the appendix we briefly describe a general context for transmutations and integral transforms used in this paper. Such a general context is formed by integral transforms composition method (ITCM).

Below we give a brief survey and outline some applications of the integral transforms composition method (ITCM) for obtaining transmutations via integral transforms. It is possible to derive wide range of transmutation operators by this method. Classical integral transforms are involved in the integral transforms composition method (ITCM) as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace and some generalized transforms. The ITCM and transmutations obtaining by it are applied to deriving connection formulas for solutions of singular differential equations and more simple non-singular ones. We consider well-known classes of singular differential equations with Bessel operators, such as classical and generalized Euler–Poisson–Darboux equation and the generalized radiation problem of A. Weinstein. Methods of this paper are applied to more general linear partial differential equations with Bessel operators, such as multivariate Bessel-type equations, GASPT (Generalized Axially Symmetric Potential Theory) equations of Weinstein, Bessel-type generalized wave equations with variable coefficients,ultra B-hyperbolic equations and others. So with many results and examples the main conclusion of this paper is illustrated: the integral transforms composition method (ITCM) of constructing transmutations is very important and effective tool also for obtaining connection formulas and explicit representations of solutions to a wide class of singular differential equations, including ones with Bessel operators.

What is ITCM and How It Works?

In transmutation theory explicit operators were derived based on different ideas and methods, often not connecting altogether. So there is an urgent need in transmutation theory to develop a general method for obtaining known and new classes of transmutations.

In this section we give such general method for constructing transmutation operators. We call this method integral transform composition method or shortly ITCM. The method is based on the representation of transmutation operators as compositions of basic integral transforms. The integral transform composition method (ITCM) gives the algorithm not only for constructing new transmutation operators, but also for all now explicitly known classes of transmutations, including Poisson, Sonine, Vekua-Erdelyi-Lowndes, Buschman-Erdelyi, Sonin-Katrakhov and Poisson-Katrakhov ones, cf. [36,37,38,39,40,41,42,43,44,45, 63,64,65] as well as the classes of elliptic, hyperbolic and parabolic transmutation operators introduced by Carroll [37,38,39].

The formal algorithm of ITCM is the next. Let us take as input a pair of arbitrary operators A, B, and also connecting with them generalized Fourier transforms F _A, F _B, which are invertible and act by the formulas

$$\displaystyle \begin{aligned} F_A A =g(t) F_A,\quad F_B B= g(t) F_B, \end{aligned} $$

(A.1)

where t is a dual variable, g is an arbitrary function with suitable properties. It is often convenient to choose g(t) = −t ² or g(t) = −t ^α, $\alpha \in \mathbb {R}$.

Then the essence of ITCM is to obtain formally a pair of transmutation operators P and S as the method output by the next formulas:

$$\displaystyle \begin{aligned} S=F^{-1}_B \frac{1}{w(t)} F_A,\quad P=F^{-1}_A w(t) F_B \end{aligned} $$

(A.2)

with arbitrary function w(t). When P and S are transmutation operators intertwining A and B:

$$\displaystyle \begin{aligned} SA=BS,\quad PB=AP. \end{aligned} $$

(A.3)

A formal checking of (A.3) can be obtained by direct substitution. The main difficulty is the calculation of compositions (A.2) in an explicit integral form, as well as the choice of domains of operators P and S.

Let us list the main advantages of Integral Transform Composition Method (ITCM).

• Simplicity—many classes of transmutations are obtained by explicit formulas from elementary basic blocks, which are classical integral transforms.

• ITCM gives by a unified approach all previously explicitly known classes of transmutations.

• ITCM gives by a unified approach many new classes of transmutations for different operators.

• ITCM gives a unified approach to obtain both direct and inverse transmutations in the same composition form.

• ITCM directly leads to estimates of norms of direct and inverse transmutations using known norm estimates for classical integral transforms on different functional spaces.

• ITCM directly leads to connection formulas for solutions to perturbed and unperturbed differential equations.

An obstacle for applying ITCM is the next one: we know acting of classical integral transforms usually on standard spaces like L ₂, L _p, C ^k, variable exponent Lebesgue spaces [46] and so on. But for application of transmutations to differential equations we usually need some more conditions hold, say at zero or at infinity. For these problems we may first construct a transmutation by ITCM and then expand it to the needed functional classes.

Let us stress that formulas of the type (A.2) of course are not new for integral transforms and its applications to differential equations. But ITCM is new when applied to transmutation theory! In other fields of integral transforms and connected differential equations theory compositions (A.2) for the choice of classical Fourier transform leads to famous pseudo-differential operators with symbol function w(t). For the choice of the classical Fourier transform and the function w(t) = (±it)^−s we obtain fractional integrals on the whole real axis, for w(t) = |x|^−s we obtain M.Riesz potential, for w(t) = (1 + t ²)^−s in formulas (A.2) we obtain Bessel potential and for w(t) = (1 ± it)^−s - modified Bessel potentials [3].

The next choice for ITCM algorithm,

$$\displaystyle \begin{aligned} A=B=B_\nu, \quad F_A=F_B=H_\nu, \quad g(t)=-t^2, \quad w(t)=j_\nu(st) \end{aligned} $$

(A.4)

leads to generalized translation operators of Delsart [47,48,49], for this case we have to choose in ITCM algorithm defined by (A.1)–(A.2) the above values (A.4) in which B _ν is the Bessel operator, H _ν is the Hankel transform, j _ν is the normalized (or “small”) Bessel function. In the same manner other families of operators commuting with a given one may be obtained by ITCM for the choice A = B, F _A = F _B with arbitrary functions g(t), w(t) (generalized translation commutes with the Bessel operator). In case of the choice of differential operator A as quantum oscillator and connected integral transform F _A as fractional or quadratic Fourier transform [50] we may obtain by ITCM transmutations also for this case [43]. It is possible to apply ITCM instead of classical approaches for obtaining fractional powers of Bessel operators [43, 51,52,53,54].

Direct applications of ITCM to multidimensional differential operators are obvious, in this case t is a vector and g(t), w(t) are vector functions in (A.1)–(A.2). Unfortunately for this case we know and may derive some new explicit transmutations just for simple special cases. But among them are well-known and interesting classes of potentials. In case of using ITCM by (A.1)–(A.2) with Fourier transform and w(t)—positive definite quadratic form we come to elliptic Riesz potentials [3, 55]; with w(t)—indefinite quadratic form we come to hyperbolic Riesz potentials [3, 55, 56]; with w(x, t) = (|x|² − it)^−α∕2 we come to parabolic potentials [3]. In case of using ITCM by (A.1)–(A.2) with Hankel transform and w(t) - quadratic form we come to elliptic Riesz B-potentials [57, 58] or hyperbolic Riesz B-potentials [59]. For all above mentioned potentials we need to use distribution theory and consider for ITCM convolutions of distributions, for inversion of such potentials we need some cutting and approximation procedures, cf. [56, 59]. For this class of problems it is appropriate to use Schwartz or/and Lizorkin spaces for probe functions and dual spaces for distributions.

So we may conclude that the method we consider in the paper for obtaining transmutations—ITCM is effective, it is connected to many known methods and problems, it gives all known classes of explicit transmutations and works as a tool to construct new classes of transmutations. Application of ITCM needs the next three steps.

Step 1.:

For a given pair of operators A, B and connected generalized Fourier transforms F _A, F _B define and calculate a pair of transmutations P, S by basic formulas (A.1)–(A.2).

Step 2.:

Derive exact conditions and find classes of functions for which transmutations obtained by step 1 satisfy proper intertwining properties.

Step 3.:

Apply now correctly defined transmutations by steps 1 and 2 on proper classes of functions to deriving connection formulas for solutions of differential equations.

The next part of this article is organized as follows. First we illustrate step 1 of the above plan and apply ITCM for obtaining some new and known transmutations. For step 2 we prove a general theorem for the case of Bessel operators, it is enough to solve problems to complete strict definition of transmutations. And after that we give an example to illustrate step 3 of applying obtained by ITCM transmutations to derive formulas for solutions of a model differential equation.

Application of ITCM to Index Shift B–Hyperbolic Transmutations

In this section we apply ITCM to obtain integral representations for index shift B-hyperbolic transmutations. It corresponds to step 1 of the above plan for ITCM algorithm.

Let us look for the operator T transmuting the Bessel operator B _ν into the same operator but with another parameter B _μ. To find such a transmutation we use ITCM with Hankel transform. Applying ITCM we obtain an interesting and important family of transmutations, including index shift B-hyperbolic transmutations, “descent” operators, classical Sonine and Poisson-type transmutations, explicit integral representations for fractional powers of the Bessel operator, generalized translations of Delsart and others.

So we are looking for an operator $T_{\nu , \mu }^{(\varphi )}$ such that

$$\displaystyle \begin{aligned} T_{\nu, \mu}^{(\varphi)} B_{\nu} = B_{\mu} T_{\nu, \mu}^{(\varphi)} \end{aligned} $$

(A.5)

in the factorized due to ITCM form

$$\displaystyle \begin{aligned} T_{\nu, \mu}^{(\varphi)} = H_{\mu}^{-1} \Big( \varphi(t) H_{\nu}\Big), \end{aligned} $$

(A.6)

where H _ν is a Hankel transform. Assuming φ(t) = Ct ^α, $C\in \mathbb {R}$ does not depend on t and $T^{(\varphi )}_{\nu , \mu }=T^{(\alpha )}_{\nu , \mu }$ we can derive the following theorem.

Theorem A.1

Let f be a proper function for which the composition (A.6) is correctly defined,

$$\displaystyle \begin{aligned}\operatorname{Re}(\alpha+\mu+1)>0,\quad \operatorname{Re} \Big(\alpha+\frac{\mu-\nu}{2} \Big)<0. \end{aligned}$$

Then for transmutation operator $T^{(\alpha )}_{\nu ,\mu }$ obtained by ITCM and such that

$$\displaystyle \begin{aligned}T^{(\alpha)}_{\nu,\mu} B_{\nu} = B_{\mu} T^{(\alpha)}_{\nu,\mu} \end{aligned}$$

the next integral representation is true

$$\displaystyle \begin{aligned} \begin{aligned} &\Big( T^{(\alpha)}_{\nu,\mu} f\Big)(x) \\ &=C \frac{2^{\alpha+3}\Gamma(\frac{\alpha+\mu+1}{2})} {\Gamma(\frac{\mu+1}{2})} \Big[\frac{x^{-1-\mu-\alpha}} {\Gamma(-\frac{\alpha}{2})} \\ &\quad \times \int_{0}^{x}f(y) {{}_2F_1}\Big( \frac{\alpha+\mu+1}{2}, \frac{\alpha}{2}+1; \frac{\nu+1}{2}; \frac{y^2}{x^2}\Big)y^\nu dy +\frac{\Gamma(\frac{\nu+1}{2}) }{\Gamma(\frac{\mu+1}{2})\Gamma(\frac{\nu-\mu-\alpha}{2})} \\ &\quad \times \int_{x}^{\infty }f(y) {}_2F_1\Big( \frac{\alpha+\mu+1}{2}, \frac{\alpha+\mu-\nu}{2}+1; \frac{\mu+1}{2}; \frac{x^2}{y^2}\Big)y^{\nu-\mu-\alpha-1}dy\Big]. \end{aligned} \end{aligned} $$

(A.7)

where ₂ F ₁ is the Gauss hypergeometric function.

Corollary A.1

Let f ∈ L ²(0, ∞), α = −μ; ν = 0. For μ > 0 we obtain the operator

$$\displaystyle \begin{aligned} \Big( T^{(-\mu)}_{0,\mu} f\Big)(x) =\frac{2\Gamma(\frac{\mu+1}{2})}{\sqrt{\pi} \Gamma(\mu/2)} x^{1-\mu} \int_{0}^{x}f(y) (x^2- y^2)^{\frac{\mu}{2}-1}dy, \end{aligned} $$

(A.8)

such that

$$\displaystyle \begin{aligned} T_{0, \mu}^{(-\mu)} D^2 = B_{\mu} T_{0, \mu}^{(-\mu)} \end{aligned} $$

(A.9)

and $T^{(-\mu )}_{0,\mu }1=1$ ,

The operator (A.8) is the well-known Poisson operator (see [47]). We will use conventional symbol

$$\displaystyle \begin{gathered} {} \mathcal{P}^\mu_x f(x)=C(\mu)x^{1-\mu}\,\int_{0}^{x}f(y) (x^2- y^2)^{\frac{\mu}{2} -1}dy, \\ \mathcal{P}^\mu_x 1=1, \quad C(\mu)=\frac{2\Gamma(\frac{\mu+1}{2})}{\sqrt{\pi} \Gamma(\frac{\mu}{2})}. \end{gathered} $$

(A.10)

We remark that if u = u(x, t), $x,t\in \mathbb {R}$, u(x, 0) = f(x) and u _t(x, 0) = 0, then

$$\displaystyle \begin{aligned} \mathcal{P}^\mu_t u(x,t)|{}_{t=0}=f(x),\quad \frac{\partial}{\partial t} \mathcal{P}^\mu_t u(x,t)\Big|{}_{t=0}=0. \end{aligned} $$

(A.11)

Indeed, we have

$$\displaystyle \begin{aligned} \mathcal{P}^\mu_t u(x,t)|{}_{t=0} &=C(\mu)t^{1-\mu}\,\int_{0}^{t}u(x,y) (t^2- y^2)^{\frac{\mu}{2}-1}dy\Big|{}_{t=0} \\ &=C(\mu)\,\int_{0}^{1}u(x,ty)|{}_{t=0} (1- y^2)^{\frac{\mu}{2}-1}dy=f(x) \end{aligned} $$

and

$$\displaystyle \begin{aligned}\frac{\partial}{\partial t} \mathcal{P}^\mu_t u(x,t)\Big|{}_{t=0} =C(\mu)\,\int_{0}^{1}u_t(x,ty)|{}_{t=0} (1- y^2)^{\frac{\mu}{2}-1}dy=0. \end{aligned}$$

Corollary A.2

For f ∈ L ²(0, ∞), α=ν−μ; $-1< \operatorname {Re} \nu < \operatorname {Re} \mu$ we obtain the first “descent” operator

$$\displaystyle \begin{aligned} \Big( T^{(\nu-\mu)}_{\nu,\mu} f\Big)(x) =\frac{2\Gamma(\frac{\mu+1}{2})}{\Gamma(\frac{\mu{-}\nu}{2}) \Gamma(\frac{\nu+1}{2})} x^{1-\mu} \int_{0}^{x}f(y) (x^2-y^2)^{\frac{\mu-\nu}{2}-1}y^\nu dy. \end{aligned} $$

(A.12)

such that

$$\displaystyle \begin{aligned}T^{(\nu-\mu)}_{\nu,\mu} B_{\nu} = B_{\mu} T^{(\nu-\mu)}_{\nu,\mu},\quad T^{(\nu-\mu)}_{\nu,\mu}1 =1. \end{aligned}$$

Corollary A.3

Let f ∈ L _1,w with the weight function $w(y)=|y|{ }^{\operatorname {Re}\nu - \operatorname {Re}\mu }$, α = 0, $-1<\operatorname {Re} \mu < \operatorname {Re} \nu$. In this case we obtain the second “descent” operator:

$$\displaystyle \begin{aligned} \Big( T^{(0)}_{\nu,\mu} f\Big)(x) =\frac{2\Gamma(\nu-\mu)} {\Gamma^2(\frac{\nu-\mu}{2})} \int_{x}^{\infty }f(y) (y^2-x^2)^{\frac{\nu-\mu}{2}-1}y\,dy. \end{aligned} $$

(A.13)

In [44] the formula (A.13) was obtained as a particular case of Buschman-Erdelyi operator of the third kind but with different constant:

$$\displaystyle \begin{aligned} \Big( T^{(0)}_{\nu,\mu} f\Big)(x) =\frac{2^{1-\frac{\nu-\mu}{2}}}{\Gamma(\frac{\nu-\mu}{2})} \int_{x}^{\infty }f(y)y \left(y^2- x^2\right)^{\frac{\nu-\mu}{2}-1}dy. \end{aligned} $$

(A.14)

As might be seen in the form (A.13) as well as (A.14) the operator $T^{(0)}_{\nu ,\mu }$ does not depend on the values ν and μ but only on the difference between ν and μ.

Corollary A.4

Let f ∈ L ²(0, ∞), $\operatorname {Re}(\alpha +\nu +1)>0$, $\operatorname {Re}\alpha <0$. If we take μ = ν in (A.7) we obtain the operator

$$\displaystyle \begin{aligned} \begin{aligned} \Big( T^{(\alpha)}_{\nu, \nu} f\Big)(x) &=\frac{2^{\alpha+3}\Gamma(\frac{\alpha+\nu+1}{2})} {\Gamma(-\frac{\alpha}{2})\Gamma(\frac{\nu+1}{2})} \Big[x^{-1-\nu-\alpha} \,\int_{0}^{x}f(y) \\ &\quad \times {{}_2F_1}\Big( \frac{\alpha+\nu+1}{2}, \frac{\alpha}{2}+1; \frac{\nu+1}{2}; \frac{y^2}{x^2}\Big)y^\nu dy\\ &\quad +\int_{x}^{\infty }f(y) {{}_2F_1} \Big( \frac{\alpha+\nu+1}{2}, \frac{\alpha}{2}+1; \frac{\nu+1}{2}; \frac{x^2}{y^2}\Big)y^{-\alpha-1}dy\Big] \end{aligned} \end{aligned} $$

(A.15)

which is an explicit integral representation of the negative fractional power α of the Bessel operator: $B^\alpha _\nu$.

So it is possible and easy to obtain fractional powers of the Bessel operator by ITCM. For different approaches to fractional powers of the Bessel operator and its explicit integral representations cf. [9, 43, 51,52,53,54, 60,61,62].

Theorem A.2

If we apply ITCM with $\varphi (t) = j_{\frac {\nu -1}{2}}(zt)$ in (A.6) and with μ = ν then the operator

$$\displaystyle \begin{aligned} \begin{aligned} &\Big( T^{(\varphi)}_{\nu,\nu} f\Big) (x) \\ &=\,^\nu T_x^zf(x)=H_{\nu}^{-1} \big[ j_{\frac{\nu-1}{2}}(z t) H_{\nu}[f](t)\big](x) \\ &=\frac{2^\nu\Gamma(\frac{\nu+1}{2}) }{\sqrt{\pi}(4xz)^{\nu-1} \Gamma(\frac{\nu}{2}) }\int_{|x-z|}^{x+z}f(y) y [(z^2-(x-y)^2)((x+y)^2-z^2)]^{\frac{\nu}{2} -1} dy \end{aligned} \end{aligned} $$

(A.16)

coincides with the generalized translation operator (see [ 47 – 49 ]), for which the next properties are valid

$$\displaystyle \begin{gathered} {} \,^\nu T_x^z (B_\nu)_x= (B_\nu)_z\,^\nu T_x^z, \end{gathered} $$

(A.17)

$$\displaystyle \begin{gathered} {} \,^\nu T_x^zf(x)|{}_{z=0}=f(x),\quad \frac{\partial}{\partial z}\,^\nu T_x^zf(x)\Big|{}_{z=0}=0. \end{gathered} $$

(A.18)

More frequently used representation of generalized translation operator $\,^\nu T_z^x$ is (see [47,48,49])

$$\displaystyle \begin{gathered} {} \,^\nu T^z_xf(x)=C(\nu)\int_0^\pi f(\sqrt{x^2+z^2-2xz\cos{\varphi}})\sin^{\nu-1}{\varphi}d\varphi, \\ C(\nu)=\Big(\int_0^\pi\sin^{\nu-1}{\varphi}d\varphi\Big)^{-1}= \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\pi}\,\,\Gamma(\frac{\nu}{2})}. \end{gathered} $$

(A.19)

It is easy to see that it is the same as ours.

So it is possible and easy to obtain generalized translation operators by ITCM, and its basic properties follows immediately from ITCM integral representation.

Application of Transmutations Obtained by ITCM to Integral Representations of Solutions to Hyperbolic Equations with Bessel Operators

Let us solve the problem of obtaining transmutations by ITCM (step 1) and justify integral representation and proper function classes for it (step 2). Now consider applications of these transmutations to integral representations of solutions to hyperbolic equations with Bessel operators (step 3). For simplicity we consider model equations, for them integral representations of solutions are mostly known. More complex problems need more detailed and spacious calculations. But even for these model problems considered below application of the transmutation method based on ITCM is new, it allows more unified and simplified approach to hyperbolic equations with Bessel operators of EPD/GEPD types.

Standard approach for solving differential equations is to find its general solution first, and then substitute given functions to find particular solutions. Here we will show how to obtain general solution of EPD type equation using transmutation operators.

Proposition A.3

A general solution of the equation

$$\displaystyle \begin{aligned} \frac{\partial^2 u}{\partial x^2}=(B_\mu)_t u,\quad u=u(x,t;\mu) \end{aligned} $$

(A.20)

for 0 < μ < 1 is represented in the form

$$\displaystyle \begin{aligned} u=\int_{0}^{1}\frac{\Phi(x+t(2p-1))} {(p(1-p))^{1-\frac{\mu}{2}}} \,dp+t^{1-\mu}\int_{0}^{1}\frac{\Psi(x+t(2p-1))} {(p(1-p))^{\mu/2}}\,dp, \end{aligned} $$

(A.21)

with a pair of arbitrary functions Φ,  Ψ.

Proof

First, we consider the wave equation when a = 1,

$$\displaystyle \begin{aligned} \frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}. \end{aligned} $$

(A.22)

A general solution to this equation has the form

$$\displaystyle \begin{aligned} F(x+t)+G(x-t), \end{aligned} $$

(A.23)

where F and G are arbitrary functions. Applying operator (A.10) (obtained by ITCM) by variable t we obtain that one solution to the Eq. (A.20) is

$$\displaystyle \begin{aligned}u_1=2C(\mu)\frac{1}{t^{\mu-1}} \int_0^{t}[F(x+z)+G(x-z)] (t^2-z^2)^{\frac{\mu}{2}-1}\,dz. \end{aligned}$$

Let us transform the resulting general solution as follows

$$\displaystyle \begin{aligned}u_1=\frac{C(\mu)}{t^{\mu-1}} \int_{-t}^{t}\frac{F(x+z)+F(x-z)+G(x+z)+G(x-z)}{(t^2-z^2)^{1-\frac{\mu}{2}}} \,dz. \end{aligned}$$

Introducing a new variable p by formula z = t(2p − 1) we obtain

$$\displaystyle \begin{aligned}u_1=\int_{0}^{1}\frac{\Phi(x+t(2p-1))} {(p(1-p))^{1-\frac{\mu}{2}}} \,dp, \end{aligned}$$

where

$$\displaystyle \begin{aligned}\Phi(x+z){=}\left[F(x+z){+}F(x-z){+}G(x+z){+}G(x-z)\right] \end{aligned}$$

is an arbitrary function.

It is easy to see that if u(x, t;μ) is a solution of (A.20) then t ^1−μu(x, t;2−μ) is also a solution. Therefore the second solution to (A.20) is

$$\displaystyle \begin{aligned}u_2=t^{1-\mu}\int_{0}^{1}\frac{\Psi(x+t(2p-1))} {(p(1-p))^{\mu/2}} \,dp, \end{aligned}$$

where Ψ is an arbitrary function, not coinciding with Φ. Summing u ₁ and u ₂ we obtain general solution to (A.20) of the form (A.21). From the (A.21) we can see that for summable functions Φ and Ψ such a solution exists for 0 < μ < 1. □

Now we derive a general solution to GEPD type equation by transmutation method.

Proposition A.4

A general solution to the equation

$$\displaystyle \begin{aligned} (B_\nu)_x u=(B_\mu)_t u,\quad u=u(x,t;\nu,\mu) \end{aligned} $$

(A.24)

for 0 < μ < 1, 0 < ν < 1 is

$$\displaystyle \begin{aligned} \begin{aligned} u&=\frac{2\Gamma(\frac{\nu+1}{2})}{\sqrt{\pi} \Gamma(\frac{\nu}{2})}\Big(x^{1-\nu}\,\int_{0}^{x}(x^2- y^2)^{\frac{\nu}{2}-1}dy \int_{0}^{1}\frac{\Phi(y+t(2p-1))} {(p(1-p))^{1-\frac{\mu}{2}}} \,dp\\ &\quad +t^{1-\mu}x^{1-\nu}\,\int_{0}^{x}(x^2- y^2)^{\frac{\nu}{2}-1}dy \int_{0}^{1}\frac{\Psi(y+t(2p-1))} {(p(1-p))^{\mu/2}} \,dp.\Big) \end{aligned} \end{aligned} $$

(A.25)

Proof

Applying the Poisson operator (A.10) (again obtained by ITCM) with index ν by variable x to the (A.21) we derive general solution (A.25) to the Eq. (A.24). □

Now let apply transmutations for finding general solution to GEPD type equation with spectral parameter.

Proposition A.5

A general solution to the equation

$$\displaystyle \begin{aligned} (B_\nu)_x u=(B_\mu)_t u+b^2u,\quad u=u(x,t;\nu,\mu) \end{aligned} $$

(A.26)

for 0 < μ < 1, 0 < ν < 1 is

$$\displaystyle \begin{aligned} \begin{aligned} u&= \frac{2\Gamma(\frac{\nu+1}{2})}{\sqrt{\pi} \Gamma(\frac{\nu}{2})}\Big(x^{1-\nu}\,\int_{0}^{x}(x^2- y^2)^{\frac{\nu}{2}-1}dy \\ &\quad \times \int_{0}^{1}\frac{\Phi(y+t(2p-1))} {(p(1-p))^{1-\frac{\mu}{2}}}j_{\frac{\mu}{2}-1}(2bt\sqrt{p(1-p)}) \,dp \\ &\quad +t^{1-\mu}x^{1-\nu}\,\int_{0}^{x}(x^2- y^2)^{\frac{\nu}{2}-1}dy \\ &\quad \times \int_{0}^{1}\frac{\Psi(y+t(2p-1))} {(p(1-p))^{\mu/2}}j_{-\frac{\mu}{2}}(2bt\sqrt{p(1-p)}) \,dp.\Big) \end{aligned} \end{aligned} $$

(A.27)

Proof

A general solution to the equation

$$\displaystyle \begin{aligned}\frac{\partial^2 u}{\partial x^2}=(B_\mu)_tu+b^2u,\quad u=u(x,t;\mu),\quad 0<\mu<1 \end{aligned}$$

is (see [24, p. 328])

$$\displaystyle \begin{aligned} u&=\int_{0}^{1}\frac{\Phi(x+t(2p-1))} {(p(1-p))^{1-\frac{\mu}{2}}}j_{\frac{\mu}{2}-1}(2bt\sqrt{p(1-p)}) \,dp \\ &\quad +t^{1-\mu}\int_{0}^{1}\frac{\Psi(x+t(2p-1))} {(p(1-p))^{\mu/2}}j_{-\frac{\mu}{2}}(2bt\sqrt{p(1-p)}) \,dp. \end{aligned} $$

Applying Poisson operator (A.10) (again obtained by ITCM) with index ν by variable x to the (A.21) we derive general solution (A.25) to the Eq. (A.24). □