New Laplace convolution integrals involving exponential, error, and parabolic cylinder functions with applications in heat transfer and linear viscoelasticity

Technical Summary: New Laplace Convolution Integrals Involving Exponential, Error, and Parabolic Cylinder Functions

Problem Statement
While classical compilations of Laplace and inverse Laplace transforms are extensive, there remains a need for new integral identities that are not present in standard tables. Specifically, this paper addresses the computation of Laplace convolution integrals involving exponential functions, error functions (and their variants), and parabolic cylinder functions. These integrals frequently arise in applied physics, particularly in heat transfer problems (via Duhamel's principle) and linear viscoelasticity (via the Boltzmann superposition principle). The authors note that while the convolution theorem has been used to find inverse transforms from known integrals, the reciprocal approach—deriving new integrals from known inverse transforms—has not been systematically exploited for these specific function classes.

Methodology
The primary methodology relies on the Laplace Convolution Theorem. The authors utilize the property that the inverse Laplace transform of a product of two functions, $F(s)G(s)$, is the convolution of their respective inverse transforms, $f(t)$ and $g(t)$.

The process involves:

1. Selection of Known Inverse Transforms: Identifying pairs of functions whose inverse Laplace transforms are known (often involving exponential, error, or parabolic cylinder functions) from standard tables.
2. Application of the Convolution Theorem: Computing the convolution integral $\int_0^t f(\tau)g(t-\tau)d\tau$ by equating it to the inverse transform of the product $F(s)G(s)$.
3. Extension via Calculus Operations:
   • Repeated Integration: Using the formula for repeated integration to generalize results (e.g., deriving integrals with higher powers in the denominator).
   • Parametric Differentiation: Applying differential operators with respect to parameters (e.g., $a$ and $b$) to generate more complex integrals involving Hermite polynomials and higher-order terms.
4. Alternative Derivations: Where possible, the authors provide alternative proofs using series expansions, beta integrals, and properties of special functions (such as the Tricomi function and generalized hypergeometric functions) to verify results without relying solely on the Laplace transform.

Key Contributions and Results
The paper presents a series of theorems and corollaries establishing new convolution integrals. The main results include:

• Exponential Functions (Theorem 5 & 8): Derivation of integrals involving products of exponentials and square roots, resulting in expressions with the complementary error function ($\text{erfc}$) and Macdonald functions ($K_\nu$). Theorem 8 generalizes this to higher powers using differential operators, yielding results in terms of Hermite polynomials.
• Parabolic Cylinder Functions (Theorem 13 & 17): Computation of convolutions involving parabolic cylinder functions $D_\nu(z)$ and exponentials. These results are extended to include the error function and repeated integration, providing closed-form solutions for integrals involving $D_\nu$ and $D_{\nu-1}$.
• Error Functions (Theorem 20 & 23): Derivation of integrals involving products of exponential functions, error functions ($\text{erf}$, $\text{erfc}$, $\text{erfi}$), and Dawson integrals. These lead to new integral representations for the error and imaginary error functions.
• Generalized Hypergeometric Functions (Theorem 11 & 26): Establishment of convolution integrals involving the exponential integral and confluent hypergeometric functions, resulting in expressions containing generalized hypergeometric functions (${}_pF_q$).

Applications
The paper demonstrates the utility of these new integrals by applying them to specific physical problems:

• Heat Transfer:
  • Semi-infinite media: Solving heat conduction problems with prescribed heat fluxes (e.g., $q(t) \propto t^{-1/2}e^{-a^2/t}$ or $q(t) \propto \text{erfc}(b/\sqrt{t})$) and prescribed surface temperatures.
  • Complex boundary conditions: Addressing problems involving convective fluxes, power-law heat fluxes, and heat fluxes coupled with lateral heat loss (cooling coefficients).
  • Specific scenarios: Modeling heat diffusion from a laser pulse in an insulating material and heat supply through an expanding contact area.
• Linear Viscoelasticity:
  • Applying the results to the Becker model of creep compliance. The paper derives the strain response of a viscoelastic material subjected to a power-law stress, expressing the solution in terms of generalized hypergeometric functions.

Significance and Claims
The authors claim that the derived convolution integrals are not reported in the most common tables of integrals (citing references like [25, 28, 7]) and are not computable using standard symbolic integration engines like Mathematica.

The paper positions the Laplace convolution theorem as a powerful and straightforward tool for generating these complex integrals, noting that alternative proofs (using series or direct integration) are "highly non-trivial." As a by-product of these derivations, the authors obtain new integral representations for the error function, the complementary error function, and the imaginary error function. The work is presented as the second in a series, following a previous paper on trigonometric and hyperbolic functions, aiming to systematically expand the available analytical tools for heat transfer and viscoelasticity problems.