A fourth-order accurate numerical scheme for distributed-order Riesz space fractional diffusion equations involving the time-fractional regularized Caputo–Prabhakar derivative

In this paper, we examine what remains the same between order convergence and unbounded order convergence, as well as between unbounded order continuity and strongly unbounded order continuity. In \cite{gao2}, Gao et al. obtained the result that a sublattice of a Riesz space is order closed if and only if it is unbounded order closed. It is shown that $\sigma$-ideals and unbounded $\sigma$-ideals are the same. Additionally, it is established that injective band operators are unbounded order continuous, while bijective order bounded disjoint preserving operators are order continuous. Let $G$ be an order dense majorizing Riesz subspace of a Riesz space $E$, and let $F$ be a Dedekind complete Riesz space. In reference \cite{turan2}, the question is posed: If $T : G\rightarrow F$ is a positive strongly unbounded order continuous operator, does $T$ have a unique positive strongly unbounded order continuous extension to all of $E$? We prove that this problem has a positive answer whenever $G$ is $suo$-convergence reducing of $E$, namely, if $ x_\alpha \overset{suo}{\rightarrow} 0$ in $E$ then $x_\alpha \overset{uo}{\rightarrow} 0$ in $G$ for any net $(x_\alpha)$ in $G$.

ABSTRACT This paper presents a novel numerical scheme for solving a two‐dimensional Euler–Poisson–Darboux equation incorporating a distributed‐order space‐fractional operator and a time‐fractional derivative of Bessel type. To approximate the distributed‐order Riesz space derivative with high accuracy, the Gauss quadrature method is employed, outperforming traditional midpoint quadrature techniques. This transforms the original problem into a multiterm fractional differential equation. Spatial discretization is achieved using an alternating direction implicit (ADI) Galerkin–Legendre spectral method, while the Crank–Nicolson scheme is applied for time integration. The stability and convergence of the proposed method are rigorously analyzed and confirmed. Numerical experiments are conducted to validate the theoretical results and demonstrate the efficiency and accuracy of the approach, highlighting its potential for solving complex fractional differential equations arising in mathematical physics.

Abstract We study model-theoretic properties of a logic whose formulas take values in suitable Riesz spaces. In addition to having a set of truth values which is not linearly ordered, there is no absolute truth or falsehood value. We take inspiration from the version of continuous logic developed in [8] by Ben-Yaacov et al. and from the more general approach recently proposed in [11] by Keisler. We extend to our framework a number of results obtained by the above mentioned authors. Under suitable assumptions on the underlying Riesz space, we provide an ultraproduct construction and prove a Łoś theorem, from which compactness follows. In the framework of Riesz spaces we also address definability issues and we extend metric notions, obtaining analogues of Keisler’s pre-metric structures and pre-metric expansions of theories defined in Keisler. In particular, we prove that every theory in a countable language has a pre-metric expansion with a so-called pseudometric approximate distance.

In this paper, we study a fully discrete numerical method for approximating the solutions of a time-fractional diffusion-wave model with distributed order involving the Riesz space fractional derivative. For the discretization of the distributed-order integral, which includes the fractional Caputo operator in time, we first apply the composite midpoint quadrature rule. Then, the L 2 - 1 η method is employed to approximate the Caputo fractional derivative in time. To discretize the Riesz space fractional derivative, a fourth-order fractional-compact difference scheme based on a novel difference operator is used. We provide a thorough stability and convergence analysis of the proposed method, establishing that the convergence order is given by O ( ( Δ t ) 3 − γ + η 2 + h x 4 + h y 4 ) . Several numerical examples are presented to demonstrate the accuracy and effectiveness of the method. Additionally, we compare the proposed approach with existing numerical schemes to highlight its advantages.

A symbol-based preconditioner for a sixth-order scheme from multi-dimensional steady-state Riesz space fractional diffusion equations

The BOB-property of locally solid Riesz spaces was introduced by Labuda in 1985. In this study, we investigate certain aspects of this property. For example, it is stated that the Dedekind completion of a locally solid Riesz space E has BOB-property iff E has the same property. It is also observed that under some sufficient conditions, the property passes to the topological completion of the underlying space. We give a characterization of BOB-property of the space LbL,F, consisting of all order bounded operators from a Riesz space L into a Dedekind complete locally convex-solid Riesz space F, in terms of that of the range space F. Moreover, we introduce topologically b-order bounded operators by means of topologically b-order bounded sets and we investigate the properties of these operators. Finally, we investigate the order structure of the space of generalized b-weakly compact operators introduced by Altin-Machrafi (2022). It is for example stated under natural conditions on the domain and range spaces that whenever the range space enjoys BOB-property then the space of order bounded generalized b-weakly compact operators is stable under the modulus operation.

In an earlier paper, Buskes and the author pointed out that, given two Archimedean Riesz spaces E E and F F , it is relatively simple to construct, from their Ogasawara-Maeda representations, a third Archimedean Riesz space G G and a bi-injective Riesz bimorphism ϕ : E × F → G \phi :E\times F\to G . They further pointed out that the Riesz subspace of G G generated by ϕ ( E × F ) \phi (E\times F) is isomorphic to the Archimedean Riesz space tensor product E ⊗ ¯ F E\overline {\otimes }F constructed by Fremlin. The proof there relies on the properties of E ⊗ ¯ F E\overline {\otimes }F proved by Fremlin. In this paper we show that this approach actually gives a simple way to construct and establish the properties of E ⊗ ¯ F E\overline {\otimes }F and that any choice of G G and ϕ \phi yield isomorphic objects.

Fractional Exponential Fitting/Adapted BDF Method for Solving Riesz Space Advection-Diffusion Equation

A novel meshless approach based on new fractional spaces for solving nonlinear Riesz space distributed order reaction-diffusion equations with non-smooth solutions and stability analysis

ABSTRACTA novel fourth‐order finite difference formula coupling the Crank–Nicolson explicit linearized method is proposed to solve Riesz space fractional nonlinear reaction‐diffusion equations in two dimensions. Theoretically, under the Lipschitz assumption on the nonlinear term, the proposed high‐order scheme is proved to be unconditionally stable and convergent in the discrete ‐norm. Moreover, a ‐matrix‐based preconditioner is developed to speed up the convergence of the conjugate gradient method with an optimal convergence rate (a convergence rate independent of mesh sizes) for solving the symmetric discrete linear system. Theoretical analysis shows that the spectra of the preconditioned matrices are uniformly bounded in the open interval . This preconditioned iterative solver, to the best of our knowledge, is a new development with a mesh‐independent convergence rate for the linearized high‐order scheme. Numerical examples are given to validate the accuracy of the scheme and the effectiveness of the proposed preconditioned solver.

New insights into the Riesz space fractional variational problems and Euler–Lagrange equations

Uniqueness of a locally solid topology on a Riesz space of measurable functions

By analyzing proofs of the classical Riesz-Kantorovich theorem, the Mazón-Segura de León theorem on abstract Uryson operators and the Pliev-Ramdane theorem on C-bounded orthogonally additive operators on Riesz spaces, we find the most general (to our point of view) algebraic structure, which we call a complementary space, for which the theorem can be generalized with a similar proof. By a complementary space we mean a PO-set $G$ with a least element $0$ such that every order interval $[0,e]$ of $G$ with $e \neq 0$ is a Boolean algebra with respect to the induced order. There are natural examples of complementary spaces: Boolean rings, Riesz spaces with the lateral order. Moreover, the disjoint union of complementary spaces is a complementary space. Our main result asserts that, the set of all additive (in certain sense) functions from a complementary space to a Dedekind complete Riesz space admits a natural Dedekind complete Riesz space structure, described by formulas which are close to the classical Riesz-Kantorovich ones. This theorem generalizes the above mentioned Mazón and Segura de León and Pliev-Ramdane theorems. In the final section, we construct a model of market with an arbitrary commodity set, connected to a complementary space.

In the present paper, we expose a deep analogy of some recently discovered objects of the Riesz space theory with classical notions of Analysis such as order ideals, bands, order projections and properties like Dedekind completeness, principal projection property. A Riesz space E is said to be C-complete if every nonempty subset of fragments of an element e∈E+ has a supremum. One of our results asserts that, a Riesz space E is C-complete if and only if every lateral band of E is a projection lateral band. Another result characterizes projective lateral bands in an arbitrary Riesz space and provides an explicit formula for the corresponding lateral band projection, which was previously known for C-complete Riesz spaces. Our characterization of a projective lateral band consists of two conditions, one of which is actually a property of the space, which we call relative C-completeness, and the other one is a property of bands, which we call relative lateral bands. A similar characterization is obtained for the relative C-completeness. We describe the lateral disjoint complement A† to any subset A of E, show that A†† equals the relative lateral closure of the lateral ideal generated by A and prove that A†=A†††. Then we obtain some corollaries on the extension of orthogonally additive operators. We also provide many examples confirming the sharpness of our results.

In this paper, we have established a numerical method for a class of time-fractional and Riesz space distributed-order advection–diffusion equation with time-delay. Firstly, we transform the Riesz space distributed-order derivative term of the diffusion equation into multi-term fractional derivatives by using the Gauss quadrature formula. Secondly, we discretize time by using second-order finite differences, discretize space by using second kind Chebyshev polynomials, and convert the multi-term fractional equation to a system of algebraic equations. Finally, we solve the algebraic equations by the iterative method, and prove the stability and convergence. Moreover, relevant examples are shown to verify the validity of our algorithm.

ABSTRACT We introduce and study an axiomatic theory of V-normed U-modules, where V is a Riesz space and U is an f-algebra; the spaces U and V also have some additional structure and are required to satisfy a compatibility condition. Roughly speaking, a V-normed U-module is a module over U that is endowed with a pointwise norm operator taking values in V. The aim of our approach is to develop a unified framework, which is tailored to the differential calculus on metric measure spaces, where U and V can take many different spaces of functions.

We prove analogues of the well known infinite distributive laws for the lateral infima and suprema instead of order ones. The proofs are more involved than that for the original laws. We show that one of the laws holds true whenever both sides of the equality are well defined. The other one is false in general, even if both sides are well defined, but true for finite sets. The proofs of two laws are completely different. The question of under what assumptions on the Riesz space and objects involved in, the second distributive law is valid for infinite sets, remains unsolved.

Two fast finite difference methods for a class of variable-coefficient fractional diffusion equations with time delay

Dominable sets and Axiom of Choice-free universal completion