Fundamental Solution Research Articles (Page 1)

Abstract In this paper we investigate large-scale linear systems driven by a fractional Brownian motion (fBm) with Hurst parameter $H\in [1/2, 1)$ . We interpret these equations either in the sense of Young ( $H>1/2$ ) or Stratonovich ( $H=1/2$ ). In particular, fractional Young differential equations are well suited to modeling real-world phenomena as they capture memory effects, unlike other frameworks. Although it is very complex to solve them in high dimensions, model reduction schemes for Young or Stratonovich settings have not yet been much studied. To address this gap, we analyze important features of fundamental solutions associated with the underlying systems. We prove a weak type of semigroup property which is the foundation of studying system Gramians. From the Gramians introduced, a dominant subspace can be identified, which is shown in this paper as well. The difficulty for fractional drivers with $H>1/2$ is that there is no link between the corresponding Gramians and algebraic equations, making the computation very difficult. Therefore we further propose empirical Gramians that can be learned from simulation data. Subsequently, we introduce projection-based reduced-order models using the dominant subspace information. We point out that such projections are not always optimal for Stratonovich equations, as stability might not be preserved and since the error might be larger than expected. Therefore an improved reduced-order model is proposed for $H=1/2$ . We validate our techniques conducting numerical experiments on some large-scale stochastic differential equations driven by fBm resulting from spatial discretizations of fractional stochastic PDEs. Overall, our study provides useful insights into the applicability and effectiveness of reduced-order methods for stochastic systems with fractional noise, which can potentially aid in the development of more efficient computational strategies for practical applications.

The quasi-steady creeping flow of a viscous fluid around a slip sphere translating perpendicular to one or two large slip planar walls at arbitrary relative positions is analyzed. To solve the axisymmetric Stokes equation for the fluid flow, we construct a general solution using fundamental solutions in spherical and cylindrical coordinate systems. Boundary conditions are first applied to the planar walls using the Hankel transform and then to the particle surface using a collocation method. Numerical results of the drag force exerted by the fluid on the particle are obtained for different values of the relevant stickiness/slip and configuration parameters. Our force results agree well with existing solutions for the motion of a slip sphere perpendicular to one or two nonslip planar walls. The hydrodynamic drag force acting on the particle is a monotonic increasing function of the stickiness of the planar walls and the ratio of its radius to distance from each planar wall. With other parameters remaining constant, this drag force generally increases with increasing stickiness of the particle surface. The influence of the slip planar walls on the axisymmetric translation of a slip sphere is significantly stronger than its axisymmetric rotation.

Second-order RLC series circuits serve as fundamental models for analyzing electrical transients and steady-state sinusoidal responses in power systems, communications, and control applications. This review summarizes the use of linear differential-equation methods to describe and solve the governing second-order ordinary differential equation derived from Kirchhoff’s voltage law. We examine the three classical damping regimes underdamped, critically damped, and overdamped and show how characteristic roots and natural frequencies determine current and voltage evolution following a step or impulse excitation. Analytical solutions employing homogeneous and particular components, Laplace transforms, and phasor techniques are compared, highlighting their roles in predicting transient decay, steady-state sinusoidal behavior, and resonance phenomena. Extensions to forced inputs, quality factor evaluation, and frequency-domain interpretation are discussed, along with representative case studies illustrating the impact of component tolerances and energy loss. By consolidating fundamental theory, solution strategies, and practical considerations, this review provides a unified framework for applying linear differential-equation methods to the transient and steady-state analysis of second-order RLC series circuits.

The slow, quasi-steady, axisymmetric translational motion of a solid spherical particle within an eccentric cavity containing a hydrogel medium is analyzed using a semi-analytical approach. The hydrogel is modeled as a porous medium saturated with a microstructured fluid exhibiting micropolar behavior. No-slip and no-spin conditions are applied at both the particle surface and the cavity wall. The hydrodynamic governing equations are solved by constructing general solutions from fundamental solutions formulated in two spherical coordinate systems—one centered on the particle and the other on the cavity. A collocation method is employed to satisfy the boundary conditions at the interfaces. The obtained results show strong agreement with existing literature. This study reveals that the cavity wall, micropolar fluid characteristics, and permeability parameters significantly influence the drag force acting on the particle. These findings provide valuable insights into controlled particle motion in hydrogel environments, with potential applications in targeted drug delivery and biomedical transport systems.

Local method of fundamental solutions formulations for polyharmonic BVPs

Abstract The concepts of $\epsilon$-nets and unitary ($\delta$-approximate) $t$-designs are important and ubiquitous across quantum computation and information. Both notions are closely related and the quantitative relations between $t$, $\delta$ and $\epsilon$ find applications in areas such as (non-constructive) inverse-free Solovay-Kitaev like theorems and random quantum circuits. In recent work, quantitative relations have revealed the close connection between the two constructions, with $\epsilon$-nets functioning as unitary $\delta$-approximate $t$-designs and vice-versa, for appropriate choice of parameters. In this work we improve these results, significantly increasing the bound on the $\delta$ required for a $\delta$-approximate $t$-design to form an $\epsilon$-net from $\delta \simeq \left(\epsilon^{3/2}/d\right)^{d^2}$ to $\delta \simeq \left(\epsilon/d^{1/2}\right)^{d^2}$. We achieve this by constructing polynomial approximations to the Dirac delta using heat kernels on the projective unitary group $\mathrm{PU}(d) \cong\mathbf{U}(d)$, whose properties we studied and which may be applicable more broadly. We also outline the possible applications of our results in quantum circuit overheads, quantum complexity and black hole physics.

This paper compares the numerical performances of the boundary element method (BEM) and the method of fundamental solutions (MFS) for solving problems governed by the Laplace equation in two dimensions. They are similar methods since both use the idea of a Green fundamental solution, but they also have some essential distinctions. BEM is currently a powerful, well-known numerical technique applied to important engineering fields. BEM solves differential equations using an integral formulation that results in boundary integrations. The MFS is a meshless technique that works directly with discrete points. Nowadays, the MFS is experiencing a rediscovery due to the intensification of research on meshless techniques. It has also been used as an auxiliary tool for other methods, including BEM, modelling intermediary steps in broader, more complex discrete cases, for example, calculating fluxes and stresses from knowledge of primal variables. Aiming to understand these relevant techniques better, this article compares some characteristics of the two methods, analysing each technique’s accuracy and robustness and giving special attention to MFS particularities for solving Laplace's problems. This approach is suitable since the Laplacian is one the most sensitive differential operators within the scalar field theory, generating ill-conditioning in the resulting discrete matrix system.

For the self-adjoint operator of the pth derivative, a system of fundamental solutions is constructed. This system is analogues to the classical system of sines and cosines. The properties of such functions are studied. Classes of self-adjoint boundary conditions aredescribed. For the operator of the third derivative, the resolvent is calculated and an orthonormal basis of eigenfunctions is given.

This study presents an analysis of the fundamental solution and Green’s function in a semi-infinite orthotropic photothermoelastic medium influenced by moisture (PTM). The governing equations are reformulated in two dimensions and expressed in dimensionless form using operator theory to derive a general solution of the PTM model. Employing newly introduced harmonic functions, the fundamental solution on the boundary surface and the Green’s function within the interior region are developed from this general solution. To assess the role of moisture, numerical simulations are carried out for displacement, stress, temperature, carrier density, and moisture distribution, with results displayed graphically. Several limiting cases-namely photothermoelastic, thermoelastic, and purely elastic-are extracted from the analysis and compared with earlier findings. The outcomes provide useful insights for engineering and material science applications, particularly in semiconductor systems subjected to coupled photothermoelastic effects. The proposed fundamental and Green’s function solutions also serve as a basis for constructing analytical models in boundary element methods and for addressing defect, crack, and inclusion problems in complex materials.

Abstract The swelling behavior of lithium‐ion batteries provides critical insights into their operational status, offering significant potential for safety early‐warning systems. However, conventional flexible force sensors face substantial challenges in battery expansion monitoring due to stringent requirements regarding installation dimensions, measurement range, and long‐term durability. This study presents an innovative thin‐film force sensor based on fumed silica‐modified conductive ink, where the silica‐induced hydrogen bonding network effectively suppresses ink leveling during fabrication, thereby preserving controlled surface micro‐roughness in the printed functional layer. The optimized sensor demonstrates exceptional performance characteristics, including an extended pressure range (0–1200 kPa), rapid response time (<50 ms), and excellent cycling stability (>10 000 cycles). Integrated into a 4 × 4 sensing array, the system successfully monitors dynamic swelling force variations during the lithium iron phosphate (LFP) battery charge–discharge cycles. Furthermore, by combining voltage and swelling force data through an advanced data fusion algorithm, highly accurate state‐of‐charge estimation is achieved with 98.13% accuracy at 1% resolution, representing a significant improvement over conventional monitoring methods. This work establishes a new paradigm for battery safety management through mechanical signature analysis, providing both fundamental insights and practical solutions for next‐generation battery monitoring systems.

High-density polyethylene (HDPE) has become a preferred material for modern pipeline systems due to its exceptional advantages. Among various joining techniques, butt fusion welding is the most widely employed and cost-effective method for connecting HDPE pipes. In response to growing demands for improved weld quality and long-term reliability, this review presents a comprehensive synthesis of recent advances aimed at enhancing the overall performance of welded joints in HDPE pipelines. It begins with an examination of the intrinsic material properties of HDPE in relation to weldability, followed by an overview of the principles and procedures of butt fusion welding, and an in-depth analysis of the mechanisms and implications of welding defect formation. The review then turns to recent progress in both destructive testing (DT) and non-destructive evaluation (NDE) techniques for weld quality assessment, as well as the optimization of key welding process parameters. In addition, emerging modification strategies of HDPE material are highlighted for their potential to improve weld strength, structural stability, and overall joint integrity. Building upon these thematic areas, the review identifies existing technical barriers and unresolved knowledge gaps, and outlines future research directions aimed at advancing both fundamental understanding and practical solutions for the next generation of high-performance HDPE pipeline systems.

We investigate the Cauchy problem for a heat equation incorporating variable diffusion coefficients and fractional memory effects modeled by a separable convolution kernel. By employing the fundamental solution of the associated parabolic equation, the problem is reformulated as a Volterra-type integral equation. Under appropriate regularity assumptions, we establish existence and uniqueness of classical solutions. Furthermore, we address an inverse problem aimed at simultaneously recovering the memory kernel and the solution. Using a differentiability-based approach, we derive a stable and well-posed formulation that enables the identification of memory effects in fractional heat models.

The flexible job shop scheduling problem (FJSP) with transportation resources such as automated guided vehicles (AGVs) is prevalent in manufacturing enterprises. Multi-type AGVs are widely adopted to transfer jobs and realize the collaboration of different machines, but are often ignored in current research. Therefore, this paper addresses the FJSP with multi-type AGVs (FJSP-MTA). Considering the difficulties caused by the introduction of transportation and the NP-hard nature, the artificial bee colony (ABC) algorithm is adopted as a fundamental solution approach. Accordingly, a Q-learning hybrid multi-objective ABC (Q-HMOABC) algorithm is proposed to deal with the FJSP-MTA. First, to minimize both the makespan and total energy consumption (TEC), this paper proposes a novel mixed-integer linear programming (MILP) model. In Q-HMOABC, a three-layer encoding strategy based on operation sequence, machine assignment, and AGV dispatching with type selection is used. Moreover, during the employed bee phase, Q-learning is employed to update all individuals; during the onlooker bee phase, variable neighborhood search (VNS) is used to update nondominated solutions; and during the scout bee phase, a restart strategy is adopted. Experimental results demonstrate the effectiveness and superiority of Q-HMOABC.

In this paper, we consider the parabolic Bourguet contours and study some aspects of the Bourguet representations in fractional calculus. Some new integral representations for the gamma, psi, Mittag-Leffler and Mainardi functions are given, and the Bourguet representation of the fundamental solution of sub-diffusion equation is discussed.

Abstract The method of fundamental solutions (MFS) is known to be effective for solving 3D Laplace and Stokes Dirichlet boundary value problems in the exterior of a large collection of simple smooth objects. Here, we present new scalable MFS formulations for the corresponding elastance and mobility problems. The elastance problem computes the potentials of conductors with given net charges, while the mobility problem—crucial to rheology and complex fluid applications—computes rigid body velocities given net forces and torques on the particles. The key idea is orthogonal projection of the net charge (or forces and torques) in a rectangular variant of a “completion flow.” The proposal is compatible with one-body preconditioning, resulting in well-conditioned square linear systems amenable to fast multipole accelerated iterative solution, thus a cost linear in the particle number. For large suspensions with moderate lubrication forces, MFS sources on inner proxy-surfaces give accuracy on par with a well-resolved boundary integral formulation. Our several numerical tests include a suspension of 10,000 nearby ellipsoids, using $$2.6\times 10^7$$ 2.6 × 10 7 total preconditioned degrees of freedom, where GMRES converges to five digits of accuracy in under two hours on one workstation.

The method of fundamental solutions and a high order continuation for nonlinear bioheat transfer problems during hyperthermia treatment.

SH wave scattering in Eringen’s nonlocal elastic solid using the method of fundamental solutions

This study presents Ritz variational method for the free transverse harmonic vibration solutions of slender beams on two-parameter elastic foundations (SBo2PEFs). The studied problem is a soil-structure interaction problem of dynamics that is important in the dynamic design of foundations and buried pipelines. The domain equation is derived using variational calculus, and the total energy functional was found for harmonic vibrations in terms of the modal displacement W(x) and the derivatives W' (x), W"(x). Minimization criteria with respect to the generalized parameter of the displacement is used to find the characteristic frequency equation. The obtained Ritz equation is an eigenvalue problem. It was found that for simply supported SBo2PEF, the exact sinusoidal shape function used gave the exact eigenfrequency for any mode of vibration. For clamped-clamped SBo2PEF, a one-parameter shape function gave accurate fundamental frequency. For cantilever SBo2PEF, a one-parameter shape function gave accurate fundamental frequency solutions.

The solution of a nonhomogeneous linear Caputo fractional differential equation of order nq,(n−1)<nq<n with Caputo fractional initial conditions can be expressed using suitable Mittag–Leffler functions. In order to extend this result to such a nonhomogeneous linear Caputo fractional differential equation of order nq,(n−1)<nq<n, that also includes lower order fractional derivative terms, we can reduce such a problem to an n-system of Caputo fractional differential equations of order q,0<q<1, with corresponding initial conditions. In this work, we use an approximation method to solve the resulting system of Caputo fractional differential equations of order q with initial conditions, using the fundamental matrix solutions involving the matrix Mittag–Leffler functions. Furthermore, we compute the fundamental matrix solution using the standard eigenvalue method. This fundamental matrix solution then allows us to express the component-wise solutions of the system using initial conditions, similar to the scalar case. As a consequence, we obtain solutions to linear nonhomogeneous Caputo fractional differential equations of order nq,(n−1)<nq<n, with Caputo fractional initial conditions having lower-order Caputo derivative terms. We illustrate the method with several examples for two and three system, considering cases where the eigenvalues are real and distinct, real and repeated, or complex conjugates.

The problem of modeling the distribution of the electrostatic field of a capacitive sensor is considered. The design fea-tures and operating principle of sensors widely used in production process control systems are described. A numerical procedure for calculating the field using a combined mesh-free method based on the combined application of the fundamental solutions method and the Monte Carlo method is described. Numerical experiments have been per-formed to determine the dependence of the sensor capacitance on the ratio of the dielectric tape thickness to the gap. As part of the validation, calculations were performed using the ELCUT engineering modeling software package, which implements the finite element method. By comparing the results obtained by the combined mesh-free method and the finite element method, it is shown that the proposed approach ensures the achievement of the necessary accuracy and can be used in solving field calculation problems at the design stage of electrical devices.