A Variational Method for Conformable Fractional Equations Using Rank-One Updates
Maatank Parashar, Tejas Dhulipalla
TL;DR
This work develops a complete variational framework for conformable fractional PDEs using rank-one PGD in a Hilbert space with a symmetric coercive energy form $a(u,v)$. The method uses a greedy step that maximizes the energy Rayleigh quotient $R_N(w) = (⟨r_N,w⟩)^2 / a(w,w)$ over rank-one candidates $w = p\otimes q$, followed by an exact line search, and is accompanied by a rigorous energy decrease identity $\|u-u_{N+1}\|_a^2 = \|u-u_N\|_a^2 - \frac{⟨r_N,w_{N+1}⟩^2}{a(w_{N+1},w_{N+1})}$. An explicit weak-greedy parameter $\theta_N$ governs geometric convergence, with a uniform lower bound $\theta_*$ yielding contraction. The alternating least squares substeps are shown to be well posed (symmetric coercive problems for $p$ and $q$) and monotone in the Rayleigh quotient, and the paper provides two detailed discretizations (weighted finite elements and Grünwald-type schemes) with clear assembly, boundary-condition, and complexity analyses. Two model problems—a stationary fractional Poisson problem and a space-time fractional diffusion problem—demonstrate the approach from continuous formulation to matrix-level implementation, highlighting efficient one-dimensional energy decompositions and near-linear scaling in one-dimensional sizes. Overall, the framework delivers a scalable, variational pathway for high-dimensional conformable fractional PDEs with rigorous convergence guarantees and practical discretizations.
Abstract
We make a complete variational treatment of rank-one Proper Generalised Decomposition for separable fractional partial differential equations with conformable derivatives. The setting is Hilbertian, the energy is induced by a symmetric coercive bilinear form, and the residual is placed in the dual space. A greedy rank-one update is obtained by maximizing an energy Rayleigh quotient over the rank-one manifold, followed by an exact line search. An exact one step energy decrease identity is proved, together with geometric decay of the energy error under a weak greedy condition that measures how well the search captures the Riesz representer of the residual. The alternating least squares realization is analyzed at the level of operators, including well posedness of the alternating subproblems, a characterization of stationary points, and monotonicity of the Rayleigh quotient along the inner iteration. Discretizations based on weighted finite elements and on Grünwald type schemes are described in detail, including assembly, boundary conditions, complexity, and memory. Two model problems, a stationary fractional Poisson problem and a space time fractional diffusion problem, are treated from the continuous level down to matrices.