Preamble

This section explores the relationships between solutions to various partial differential equations, specifically focusing on the properties of the function $z(x, s; a, b)$ and its generalizations. We consider the differential operator $D_x = \frac{\partial}{\partial x}$ and its applications to sequences of functions $z(x, s; a + 2n, 0)$ and $D_x^n z(x, s; a + 4n, 0)$ for $n = 0, 1, 2, \dots$. Building upon the classical work of S. A. Chaplygin, we analyze the behavior of $z(x, s; a, b)$ and its transformations into $u[x, s; B(s)]$ and $u(x, s; a, c)$. These relationships allow for the representation of solutions to complex boundary value problems through series expansions and integral operators.

§ 1. Basic Relations

Following the methodology established in [1], we consider the function $z(x, s; a, b)$ which satisfies the following second-order partial differential equation:
$$z_{xx} = z_{ss} + \frac{a}{s} z_s + b^2 z, \quad z(x, 0) = \tau(x), \quad z_s(x, 0) = 0 \tag{1.1}$$
where $a = 2\nu > 0$. The solution can be expressed in the integral form:
$$z(x, s; a, b) = \int_{-1}^{1} \tau(x + \xi s) (1 - \xi^2)^{\nu - 1/2} T(\xi, s) d\xi \tag{1.2a}$$
In the case where $b = 0$, the solution for the parameter $a + 2n$ is denoted as $z(x, s; a + 2n, 0)$. Using the properties of the operator $D_x^n$, we derive the following relationship for the $n$-th derivative:
$$s^n D_x^n z(x, s; a + 2n, 0) = A_n \int_{-1}^{1} \tau(\xi) d\xi \tag{1.3}$$
where the coefficients $A_n$ are determined by the Gamma function $\Gamma(\nu)$ and the specific indexing of the parameter $a$.

By substituting the integral representation (1.2a) into the governing equation, we obtain a series expansion for the general case $b \neq 0$:
$$z(x, \lambda s; a_2, b) = \sum_{n=0}^{\infty} g_n(\nu) s^{2n} D_x^n z(x, s; a_1 + 2n, 0) \tag{1.9a}$$
where the coefficients $g_n(\nu)$ involve hypergeometric functions of the form:
$$g_n(\nu) = \frac{k^2 \lambda^2 s^2}{2^{2n} n! (\nu + 1)_n} F(-n, \nu_1 + 1, \nu_2 + 1; k^2 \lambda^2 s^2) \tag{1.9b}$$
This formulation allows us to relate solutions with different parameters $a_1$ and $a_2$ through the action of the spatial derivative operator.

§ 2. Integral Representations and Transformations

For the case where $b = 0$, the relationship between the initial data $\tau(x)$ and the solution $z(x, s; a, 0)$ can be inverted. Specifically, we can express $\tau(x)$ as a series:
$$\tau(x) = \sum_{n=0}^{\infty} g_n(\nu) s^{2n} D_x^n z(x, s; a + 4n, 0) \tag{1.10a}$$
Alternatively, using the operator $W = \sqrt{s^2 D_x^2 - b^2}$, we can write:
$$\tau(x) = W z(x, s; a, b) \tag{1.10c}$$
These expressions demonstrate that the solution at any $s > 0$ contains sufficient information to reconstruct the initial state $\tau(x)$ via differential operators.

When considering the transformation to the function $u(x, s; a, c)$, we utilize the Gauss hypergeometric function $F(-n, p+n, q; x)$. For $a_1 > a_2 > 0$, the following integral relation holds:
$$s^{2n} D_x^{2n} z(x, s; a_1 + 4n, 0) = \delta_n \int_{-1}^{1} (1 - \xi^2)^{\nu_1 - 1/2} G_n(\nu_1, \nu_2 + 1; \xi^2) z(x, \xi s; 0, 0) d\xi \tag{1.12}$$
where $\delta_n$ is a normalization constant involving $\Gamma(\nu_1 + 1)$ and $\Gamma(\nu_1 - \nu_2 + n)$. In the limit where $a_2 = 0$ and $b = 0$, the expression simplifies to:
$$z(x, \lambda s; 0, 0) = \sum_{n=0}^{\infty} \gamma_n(\nu) s^{2n} D_x^n z(x, s; a + 4n, 0) \tag{1.13a}$$
This allows for a direct connection between the symmetric mean of the initial data $2z(x, \lambda s; 0, 0) = \tau(x + \lambda s) + \tau(x - \lambda s)$ and the higher-order parameter solutions.

§ 3. Generalized Hypergeometric Representations

We further generalize these results by setting $\lambda = 1$ and $a_1 = a_2 = a$ in the previous relations. The function $z(x, s; a, b)$ can be expanded as:
$$z(x, s; a, b) = \sum_{n=0}^{\infty} A_n s^{2n} D_x^n z(x, s; a + 2n, 0) \tag{1.15a}$$
where the coefficients $A_n$ are related to the Bessel function $J_{\nu+2n}(bs)$ as follows:
$$A_n = \frac{(-1)^n \Gamma(\nu + n)}{n! \Gamma(\nu + 2n)} \left(\frac{bs}{2}\right)^n J_{\nu+2n}(bs) \tag{1.15b}$$
This representation is particularly useful for analyzing the asymptotic behavior of the solution as the parameter $b$ varies.

For the function $u(x, s; a, c)$, we obtain a similar expansion:
$$u(x, \lambda s; a_2, c) = \sum_{n=0}^{\infty} A_n s^{2n} D_x^n z(x, s; a_1 + 4n, 0) \tag{1.12a}$$
The coefficients here involve the generalized hypergeometric function $_3F_2$:
$$A_n = g_n(\nu_1) {}_3F_2(-n, \nu_1 + n, 1; p_2 + 1, q_2 + 1; \lambda^2) \tag{1.12b}$$
These results unify several known special cases in the theory of singular partial differential equations and provide a systematic way to construct solutions using known basis functions.

§ 4. Asymptotic Limits and Connections to Heat Equations

By performing a change of variables $s = 2\sqrt{\epsilon} s_1$ and $a = 2\epsilon - 1$, and taking the limit as $\epsilon \to \infty$, the equation (1.1) transforms into a parabolic type:
$$\lim_{\epsilon \to \infty} z(x, 2\sqrt{\epsilon} s_1; a, b) = w(x, s_1; b) \tag{4.1}$$
where $w$ satisfies the heat-like equation $w_{s_1} = w_{xx} - b^2 w$. Under this limit, the differential operators transform as:
$$\lim_{\epsilon \to \infty} D_s^n z = D_{s_1}^n w \tag{4.3}$$
This allows us to map the properties of the hyperbolic equation (1.1) onto the parabolic domain. For instance, the expansion (1.9) becomes:
$$w(x, s_1; b) = \sum_{n=0}^{\infty} \frac{A_n s_1^n}{n!} D_x^n w(x, s_1; 0) \tag{4.4a}$$
which is a known result in the theory of heat kernels.

Furthermore, the integral representation (1.2a) in this limit yields:
$$w(x, s_1; 0) = \int_{-\infty}^{\infty} \tau(x + \xi) \frac{1}{\sqrt{4\pi s_1}} e^{-\xi^2 / 4s_1} d\xi \tag{4.6}$$
This confirms that our generalized framework correctly reduces to the classical Weierstrass transform for the heat equation. Similar limits applied to the $u(x, s; a, c)$ function lead to new representations for solutions of confluent hypergeometric differential equations.

§ 5. Applications to Chaplygin-type Equations

Finally, we apply these methods to the generalized Chaplygin equation:
$$u_{ss} + B(s) u_s = u_{xx}, \quad u(x, 0) = \tau(x) \tag{5.1a}$$
where $B(s) = \frac{d}{ds} \ln \sqrt{K(s)}$. Using the integral operator (5.2), we represent the solution as:
$$u(x, s; B) = \sum_{n=0}^{\infty} A_n(s) D_x^n z(x, s; a + 4n, 0) \tag{5.3a}$$
The coefficients $A_n(s)$ are determined by a system of recurrence relations:
$$A_{n+1} = -2 \int s A_n ds + \text{const} \tag{5.6}$$
This approach allows for the construction of solutions for various gas dynamics problems where the function $K(s)$ takes specific forms, such as $K(s) = s^m$ or more complex analytic expressions. The convergence of these series and their relationship to the fundamental solutions of the associated singular equations are established through the properties of the operator $D_x^n$.

§ 6. Examples and Special Cases

To illustrate the theory, we consider the case where $\tau(x)$ satisfies the harmonic condition $\tau''(x) + \kappa^2 \tau(x) = 0$. The solutions take the form of product functions:
$$z(x, s; a, b) = \tau(x) s^{\nu-1} J_{\nu}(b_0 s), \quad w(x, s; b) = \tau(x) e^{-b_0^2 s} \tag{6.1a, b}$$
where $b_0 = \sqrt{b^2 + \kappa^2}$. For polynomial initial data $\tau(x) = x^m$, the solutions are expressed via hypergeometric functions $_2F_1$ and $_3F_2$. These examples validate the general expansion formulas derived in the preceding sections and demonstrate their utility in practical calculations.

The results presented here extend the work of previous authors [8-12] by providing a unified operator-based approach to the study of singular partial differential equations. The use of the $D_x$ operator and its associated series expansions offers a powerful tool for both theoretical analysis and the numerical approximation of solutions in mathematical physics.