Constructive solutions of the heat equation with Stieltjes derivatives

Abstract: In this work, we investigate the one-dimensional heat equation within the framework of Stieltjes calculus. We first consider the equation associated with two fixed derivators and develop a constructive approach to establish the existence of solutions. We then study the corresponding initial value problem and incorporate several types of boundary conditions. Finally, we introduce a notion of multivariable derivator, suitable for higher-dimensional settings, and obtain explicit solutions of the heat equation for relevant classes of such derivators.

• The paper develops explicit separation-of-variables solutions for heat equations with Stieltjes time and space derivatives, including jumps, flat intervals, initial data, and selected boundary conditions.
• Its convergence theorems establish when infinite modal and heat-polynomial series can be differentiated term by term, with a Widder-type radius condition ensuring valid solutions for sum derivators.
• The study shows that Fourier-style Dirichlet and Neumann methods require translation-invariant derivators, while general two-variable derivators usually destroy separability and remain an open problem.

Overview and motivation

This paper develops a constructive solution theory for the one-dimensional heat equation formulated with Stieltjes derivatives, i.e., derivatives taken with respect to left-continuous, nondecreasing functions (derivators) $g,h:\mathbb{R}\to\mathbb{R}$. The framework unifies continuous dynamics, jump phenomena encoded in the atomic part of $\mu_g$, and latency periods where $g$ is constant, thereby providing an alternative to classical impulsive parabolic models that avoids imposing jump conditions explicitly. Prior work on parabolic Stieltjes problems relied mainly on abstract functional-analytic tools—semigroups, diagonalization, Stieltjes–Bochner spaces (2605.18585)—whereas the present contribution pursues explicit representations via separation of variables, together with precise convergence conditions under which the resulting series are genuine solutions. A second contribution is the introduction of multivariable derivators $G:\mathbb{R}^n\to\mathbb{R}$, which generalize product time scales by allowing interdependence among the measures associated with different variables.

Preliminaries

The paper assembles the standard toolkit of Stieltjes calculus: the Lebesgue–Stieltjes measure $\mu_g$; the sets $D_g$ of discontinuities and $C_g$ of constancy intervals; the $g$-derivative, which at a jump point reduces to $(f(t^{+})-f(t))/\Delta g(t)$; the spaces $\mathcal{BC}_g^k$ with norm $\mu_g$0; and the Fundamental Theorem of Calculus for $\mu_g$1-absolutely continuous functions. Central objects include the $\mu_g$2-exponential

$\mu_g$3

which solves first-order linear problems even without regressivity (in which case solutions may vanish past a point where $\mu_g$4), and the $\mu_g$5-sine, cosine, hyperbolic sine and cosine defined through linear systems. The paper also recalls the theory of $\mu_g$6-monomials $\mu_g$7 and Stieltjes-analytic functions from prior work, establishing series representations such as $\mu_g$8 for $\mu_g$9. Two auxiliary results are proved: continuity of the integration operator on $g$0 under uniform limits, and continuity of the differentiation operator $g$1, the latter yielding term-by-term differentiation of convergent series.

General solution via separation of variables

For two derivators $g$2 (time) and $g$3 (space), the heat equation is

$g$4

with the technical assumption that $g$5 and $g$6 so that derivatives exist at the corner points. The main structural result characterizes nontrivial separated solutions: $g$7 solves the equation if and only if there exists $g$8 such that $g$9 and $G:\mathbb{R}^n\to\mathbb{R}$0. The temporal factor is an exponential $G:\mathbb{R}^n\to\mathbb{R}$1; the spatial factor is a combination of $G:\mathbb{R}^n\to\mathbb{R}$2-exponentials $G:\mathbb{R}^n\to\mathbb{R}$3, expressible via $G:\mathbb{R}^n\to\mathbb{R}$4-hyperbolic or trigonometric functions depending on the sign of $G:\mathbb{R}^n\to\mathbb{R}$5. Finite superpositions give explicit solutions for arbitrary finite collections $G:\mathbb{R}^n\to\mathbb{R}$6.

The infinite-series case is handled by requiring that, for each fixed $G:\mathbb{R}^n\to\mathbb{R}$7, the series $G:\mathbb{R}^n\to\mathbb{R}$8 converge in $G:\mathbb{R}^n\to\mathbb{R}$9, and that, for each fixed $\mu_g$0, $\mu_g$1 converge in $\mu_g$2. Under these hypotheses the sum is again a solution—an implication being that classical Fourier-type expansions carry over verbatim once convergence is measured in the appropriate Stieltjes spaces. The paper also provides more checkable sufficient conditions phrased in terms of pointwise and uniform convergence of the coefficient-weighted series and their derivatives.

Initial value problem

For initial data of the form

$\mu_g$3

the IVP admits the explicit solution obtained by pairing each spatial mode with its corresponding $\mu_g$4-exponential in time initialized at $\mu_g$5 or $\mu_g$6. This extends to countably infinite expansions: if the series defining $\mu_g$7 converges pointwise and the resulting modal sequences satisfy the convergence conditions of the series theorem, the infinite superposition solves the IVP. An illustrative example uses derivators with a unit jump ($\mu_g$8 for $\mu_g$9) and a flat segment ($D_g$0 constant on $D_g$1), producing the solution $D_g$2 for $D_g$3—demonstrating concretely how jumps and latency intervals propagate into the solution structure. Notably, existence here is established only for data lying in the span (or closure under these specific modes) of $D_g$4-exponentials; no general well-posedness claim comparable to the diagonalization-based results of earlier work is made.

Boundary conditions

Periodic conditions

For periodic boundary conditions $D_g$5, $D_g$6, separation of variables requires a spectral parameter $D_g$7 satisfying $D_g$8; this condition ensures the auxiliary first-order problem has a nontrivial solution, without which nontriviality of $D_g$9 cannot be asserted—a limitation stated explicitly. The construction decomposes the second-order periodic problem into two first-order problems and extends the known real-root uniqueness theorem to complex roots in this special case.

Periodicity of the derivator

To recover Dirichlet and Neumann eigenfunctions analogous to $C_g$0 and $C_g$1, the derivator must satisfy the translation-invariance condition

$C_g$2

satisfied, e.g., by affine derivators $C_g$3. Under this hypothesis the paper proves three structural facts: translation invariance of $C_g$4 on $C_g$5-measurable sets; preservation of $C_g$6 and $C_g$7 under translations by $C_g$8 (which also yields $C_g$9); and invariance of the trigonometric system under translated arguments. Combined with the monomial series representations, sufficient conditions on the $g$0-monomials at $g$1—namely vanishing of the odd-monomial series and normalization of the even-monomial series at $g$2—guarantee $g$3 and $g$4. Consequently, $g$5 solves the Dirichlet problem and its cosine counterpart solves the Neumann problem. The implication is that the classical Fourier method survives in the Stieltjes setting precisely when the derivator respects the spatial period; without condition (g_condicion) no analogue of $g$6 is guaranteed.

Heat equation with a two-variable derivator

The paper introduces $g$7-variable derivators $g$8 whose coordinate sections are left-continuous and nondecreasing, and defines partial $g$9-derivatives via quotients of increments of $(f(t^{+})-f(t))/\Delta g(t)$0 against increments of $(f(t^{+})-f(t))/\Delta g(t)$1 along each axis. When $(f(t^{+})-f(t))/\Delta g(t)$2 has nonzero classical partials, $(f(t^{+})-f(t))/\Delta g(t)$3; recovering the classical operator forces $(f(t^{+})-f(t))/\Delta g(t)$4, identifying the sum-of-derivators case with the usual setting.

Sum of one-variable derivators

For $(f(t^{+})-f(t))/\Delta g(t)$5, the $(f(t^{+})-f(t))/\Delta g(t)$6-derivatives coincide exactly with the $(f(t^{+})-f(t))/\Delta g(t)$7- and $(f(t^{+})-f(t))/\Delta g(t)$8-derivatives of Section 3, so all previous results transfer. The paper then constructs two-variable $(f(t^{+})-f(t))/\Delta g(t)$9-polynomials $\mathcal{BC}_g^k$0 recursively via commuting integration operators $\mathcal{BC}_g^k$1 and $\mathcal{BC}_g^k$2; when $\mathcal{BC}_g^k$3, they factor as $\mathcal{BC}_g^k$4, and for continuous derivators reduce to $\mathcal{BC}_g^k$5 (recovering $\mathcal{BC}_g^k$6 for identities). The central result mirrors Widder's heat polynomials: a uniformly convergent, term-by-term differentiable series $\mathcal{BC}_g^k$7 solves the $\mathcal{BC}_g^k$8-heat equation if and only if the coefficients satisfy

$\mathcal{BC}_g^k$9

which collapses the double series into heat $\mu_g$00-polynomials $\mu_g$01 satisfying $\mu_g$02 and $\mu_g$03. Convergence is governed by a radius

$\mu_g$04

with absolute convergence guaranteed wherever $\mu_g$05; since $\mu_g$06 is nondecreasing this region is either trivial, the full interval, or a prefix interval $\mu_g$07 or $\mu_g$08. Crucially, differentiated series remain heat-$\mu_g$09-polynomial series with the same radius, justifying term-by-term differentiation and yielding a complete constructive solution theorem whenever $\mu_g$10.

Product derivator

For $\mu_g$11 with positive derivators, separated solutions require $\mu_g$12 and $\mu_g$13. The time equation is solvable by the $\mu_g$14-exponential provided $\mu_g$15; the space equation has nonconstant coefficients and is treated via the general theory of second-order Stieltjes equations with variable coefficients, giving unique solvability when $\mu_g$16 and a Wronskian-type independence criterion requiring $\mu_g$17 on $\mu_g$18. An instructive remark clarifies why $\mu_g$19 does not reproduce classical partial derivatives directly: the correct object is the derivative against the variation of $\mu_g$20 in the orthogonal direction, since $\mu_g$21.

Limitations and open questions

Several restrictions bound the scope of the results. Existence for the IVP is established only for initial data expandable in the specific $\mu_g$22-exponential modes considered; general $\mu_g$23-type completeness and uniqueness results, available in the diagonalization framework of prior work, are not obtained here. The periodic boundary analysis depends on the constraint $\mu_g$24, and the Dirichlet/Neumann constructions require the strong translation-invariance condition on the derivator together with monomial-series conditions at $\mu_g$25—conditions not characterized intrinsically. For product derivators, the spatial equation's coefficients are nonconstant, so closed-form solutions are unavailable beyond existence statements. Most significantly, for a general two-variable derivator $\mu_g$26, the paper shows that separation of variables breaks down: the reduced equations acquire explicit dependence on the frozen coordinates $\mu_g$27 through chain-rule factors $\mu_g$28, and the authors state plainly that these dependencies prevent a straightforward application of the method. Establishing a solution theory for equation (Gcalor) with general $\mu_g$29, and proving existence/uniqueness results analogous to the semigroup-based literature, remain open.

Conclusion

The paper delivers an explicit, constructive treatment of the Stieltjes heat equation across three regimes: fixed pairs of one-variable derivators (with IVP and periodic/Dirichlet/Neumann boundary data), sum derivators supporting a full theory of heat $\mu_g$30-polynomials with a Widder-type convergence radius, and product derivators with variable-coefficient reductions. Its main technical contributions are the precise convergence criteria legitimizing term-by-term differentiation of solution series, the identification of derivator periodicity as the exact hypothesis restoring Fourier-type eigenfunctions, and the introduction of multivariable derivators extending product time scales. The failure of separability for general $\mu_g$31 delineates clearly where the constructive method ends and where new techniques would be required.