Parabolic Equations Research Articles

A Two-Dimensional Nonlocal Fractional Parabolic Initial Boundary Value Problem

In this paper, we investigate a two-dimensional singular fractional-order parabolic partial differential equation in the Caputo sense. The partial differential equation is supplemented with Dirichlet and weighted integral boundary conditions. By employing a functional analysis method based on operator theory techniques, we prove the existence and uniqueness of the solution to the posed nonlocal initial boundary value problem. More precisely, we establish an a priori bound for the solution from which we deduce the uniqueness of the solution. For proof of its existence, we use various density arguments.

Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation and its applications

Abstract In this paper, we study discrete Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation. We then apply these Carleman estimates to investigate two inverse problems for the space semi-discrete stochastic parabolic equations, including a discrete inverse random source problem and a discrete Cauchy problem. We firstly establish two Carleman estimates for a one-dimensional semi-discrete stochastic parabolic equation, one for homogeneous boundary and the other for non-homogeneous boundary. Then we apply these two estimates separately to derive two stability results. The first one is the Lipschitz stability for the discrete inverse random source problem. The second one is the H"{o}lder stability for the discrete Cauchy problem.

Existence and Asymptotic Behaviors to a Nonlinear Fourth-order Parabolic Equation with a General Source

Existence and Asymptotic Behaviors to a Nonlinear Fourth-order Parabolic Equation with a General Source

Shape optimization of a non-uniform piezoelectric bending beam for human knee energy harvester

Abstract Scavenging energy from the human body to provide a sustainable source for electronic devices has gained significant attention. Recently, scientists have focused on harnessing biomechanical energy from human motion. This study was dedicated to developing and optimizing a non-uniform piezoelectric bending beam-based human knee energy harvester. The bimorph non-uniform piezoelectric bending beam consisted of a non-uniform carbon fiber substrate and piezoelectric macro fiber composites. Compared to the uniform piezoelectric bending beam, the non-uniform piezoelectric beam can optimize the shape to improve the average strain, thus improving the energy harvesting efficiency. In this study, eight shape functions, including ellipse, sin, tanh, exponential function, parabola, trigonometric line, and bell curves, were investigated and optimized. The bell curve bending beam was selected and fabricated due to its good performance. Then, a benchmark platform was developed to test the deflection curve and reaction force when the nonuniform bending beam was compressed. Finally, to validate the design, experimental testing on three subjects was conducted when they were equipped with the harvester and walked on a treadmill. Testing results indicated that the non-uniform bending beam-based energy harvester can improve the energy harvesting efficiency by 28.57% compared to the uniform beam-based energy harvester. The output power can reach 18.94 mW when walking at 7.0 km h−1.

Exploring Rb2YCuCl6 and Cs2YCuCl6 double perovskites: Structural, electronic, optical, elastic, and thermoelectric properties via density functional theory

Double perovskites have gathered momentous attention due to their structural, optoelectronics, and thermal properties. FP-LAPW + lo technique was employed along with the PBEsol-GGA and TB-mBJ potential. The optimized parameters and E-V parabolic curve were computed employing Birch Murnaghan's equation of state (BM-EOS). The negative formation energy Ef for Rb2YCuCl6 and Cs2YCuCl6 confirms the thermodynamic stability of these double perovskites. The mechanical stability of both compounds was confirmed by their elastic constants Cij, Pugh's ratio, and anisotropy. The indirect band plots of Rb2YCuCl6 (2.57eV) and Cs2YCuCl6 (2.39eV) double perovskites confirm the semiconductor nature. The optical properties like dielectric function, reflectivity, optical conductivity, absorption coefficient, and energy loss function were determined within an energy range of up to 18 eV. The high absorption spectra for the under-study compounds 147.21 × 104 cm−1 at 14.60 eV for Rb2YCuCl6 and 261.30 × 104 cm−1 at 15.57 eV for Cs2YCuCl6 lies in the far UV region determines its potential use in the high-frequency devices. The thermoelectric properties such as the Seebeck coefficient, ZT, and power factor, are investigated using the semi-classical Boltzmann theory implemented in the BoltzTrap code. The peak value of ZT for Rb2YCuCl6 is 0.38 and for Cs2YCuCl6 is 0.33. The suggested findings indicate that Rb2YCuCl6 and Cs2YCuCl6 are potential candidates for thermoelectric and photovoltaic applications.

Additional Conditions in Boundary Value Problems of Heat Conduction (Review)

A review of studies related to the use of additional boundary conditions (ADBs) and additional sought functions (ADFs) in obtaining analytical solutions to heat conduction problems is presented. ADВs allow the equation to be executed at the boundaries, which leads to its execution inside the domain, excluding direct integration over the spatial coordinate. ADF allows one to reduce a partial differential equation to an ordinary differential equation, from the solution of which the eigenvalues of the boundary value problem are found. Eigenvalues in classical methods are found from the solution of the Sturm–Liouville boundary value problem formulated in the domain of a spatial variable. Consequently, the method used in this work leads to another algorithm for their determination, based on the solution of a temporary differential equation, the order of which is determined by the number of approximations of the resulting solution. In a problem based on determining the front of a temperature disturbance, the equivalence of solutions to the parabolic and hyperbolic heat equations was found. And, in particular, a number of approximations have been found that limit the speed of propagation of a thermal wave in the solution of a parabolic equation to a value equal to its real value for a specific material, at which it coincides with the solution of the hyperbolic equation.

DC current sensor based upon a fused fiber optic Fabry-Perot interferometer (FPI) and copper wire

In order to meet the demand for direct current (DC) current measurements in industrial production, a new type of fiber optic DC current sensor, combining thin copper wire and fiber optic Fabry-Perot interferometer (FPI), is reported. FPI is a sandwich structure formed by splicing a single-mode fiber (SMF) and a quartz capillary, with a cavity length of approximately 200 μm. The copper wire, with a length of 3.2 cm and a diameter of 1 mm, is attached to the FPI to form the sensor. Due to the excellent thermal conductivity of copper, the results showed that the sensor increased the sensitivity from 4.1 pm/°C to 11 pm/°C. After applying direct current to the sensor, a large quantity of heat generated by the copper wire was absorbed by the FPI, resulting in a significant shift in the resonant wavelength of FPI, which can indirectly measure the current. Three current experiments were performed, and parabolic curves have been used in fitting of the Dip wavelength and the square of the current. The results provide high correlation coefficients with values exceeding 0.99. Therefore, this curve may be used to measure the square of the current, further obtaining the current. In summary, the sensor is compact and durable, easy to manufacture, and provides high correlation coefficients, providing an alternative for measuring DC current.

Stability radius and dichotomy radius of infinite-dimensional linear systems under unbounded perturbations via Yosida distance

In this paper, by using the concept of Yosida distance between two closed linear operators, we study the stability radius r(A) of linear systems u′(t)=Au(t), where A is the generator of an analytic semigroup, under unbounded perturbations in this class of generators. We show that r(A)=1/sups∈R‖R(is,A)‖L(X), so extending a classic result by Henrichsen and Pritchard to the infinite-dimensional case. A formula of the dichotomy radius is also established. Finally we give an estimate of the stability radius of general C0-semigroups. Two examples from a parabolic equation are given.

Predator–prey systems with a variable habitat for predators in advective environments

AbstractCommunity composition in aquatic environments can be shaped by a broad array of factors, encompassing habitat conditions in addition to abiotic conditions and biotic interactions. This paper pertains to reaction–diffusion–advection predator–prey model featuring a variable predator habitat in advective environments, governed by a unidirectional flow. First, we establish the near‐complete global dynamics of the system. In instances where the functional response to predation conforms to Holling‐type I or II, we explore the uniqueness and stability of positive steady‐state solutions via the application of particular auxiliary techniques, the comparison principle for parabolic equations, and perturbation analysis. Furthermore, we obtain the critical positions at the upper and lower boundaries of the predator's habitat, which determine the survival of the prey irrespective of the predator's growth rate. Finally, we show how the location and length of the predator's habitat affect the persistence and extinction of predators and prey in the event of a small population loss rate at the downstream end. From the biological point of view, these results contribute to our deeper understanding of the effects of habitat on aquatic populations and may have applications in aquaculture and the establishment of protection zones for aquatic species.

Some qualitative analysis for a parabolic equation with critical exponential nonlinearity

Some qualitative analysis for a parabolic equation with critical exponential nonlinearity

A finite volume approximations for one nonlinear and nonlocal integrodifferential equations

In this paper, the error analysis of the Petrov–Galerkin finite volume element method (FVEM) is investigated for a nonlinear parabolic integro-differential equation that arises in the mathematical modeling of the penetration of a magnetic field into a substance, accounting for temperature-dependent changes in electrical conductivity. Starting from Maxwell’s equations, we derive a one-dimensional model problem, which forms the basis of our analysis. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. For simplicity, we consider only the lowest-order (linear and L-splines) finite volume elements. The novel contribution lies in the application of FVEM to this problem, leading to the establishment of an unconditionally stable numerical scheme and the derivation of optimal error estimates in the L∞(L2(Ω)) and L2(H01(Ω)) norms for both semi-discrete and linearized backward Euler fully-discrete schemes, using a generalized projection method that carefully manages the nonlinear terms. Lastly, numerical experiments are provided to support the theoretical conclusions.

An exponential spectral deferred correction method for multidimensional parabolic problems

We present some efficient algorithms based on an exponential time differencing spectral deferred correction (ETDSDC) method for multidimensional second and fourth-order parabolic problems with non-periodic boundary conditions including Dirichlet, Neumann, Robin boundary conditions. Similar to the Fourier spectral method for periodic problems, the key to the efficiency of our algorithms is to construct diagonal discrete linear operators via Legendre–Galerkin methods with Fourier-like basis functions. In combination with the ETDSDC scheme, the proposed methods are spectrally accurate in space and up to 10th-order accurate in time (as shown in this work). We demonstrate the high-order of convergence and efficiency of our algorithms in solving parabolic equations through a series of two-dimensional and three-dimensional examples including Ginzburg–Landau and Allen–Cahn equations.

On a parabolic Monge–Ampère type equation on compact almost Hermitian manifolds

AbstractWe investigate a parabolic Monge–Ampère type equation on compact almost Hermitian manifolds and derive a priori gradient and second‐order derivative estimates for solutions to this parabolic equation. These a priori estimates give us higher order estimates and a long‐time solution. Then, we can observe its behavior as .

Inverse problem of reconstructing source term for a class of non‐divergence parabolic equations

This paper explores an inverse problem pertaining to the determination of a source function in non‐divergence parabolic equations, where the solution is known at a discrete set of points. Being different from other ordinary inverse source problems, which are often dependent on only one variable, the unknown coefficient in this paper not only depends on the space variable but also depends on the time . On the basis of the optimal control framework, the existence of the optimal solution of the control function is proved. The necessary conditions to be satisfied by the optimal solution are given. The convergence of the optimal solution when the mesh parameters tend to zero is obtained. The conjugate gradient method is applied to the inverse problem and some numerical results are presented for various typical test examples.

Neural Operator Approximations for Boundary Stabilization of Cascaded Parabolic PDEs

ABSTRACTThis article proposes a novel method to accelerate the boundary feedback control design of cascaded parabolic difference equations (PDEs) through DeepONet. The backstepping method has been widely used in boundary control problems of PDE systems, but solving the backstepping kernel function can be time‐consuming. To address this, a neural operator (NO) learning scheme is leveraged for accelerating the control design of cascaded parabolic PDEs. DeepONet, a class of deep neural networks designed for approximating nonlinear operators, has shown potential for approximating PDE backstepping designs in recent studies. Specifically, we focus on approximating gain kernel PDEs for two cascaded parabolic PDEs. We utilize neural operators to map only two kernel functions, while the other two are computed using the analytical solution, thus simplifying the training process. We establish the continuity and boundedness of the kernels, and demonstrate the existence of arbitrarily close DeepONet approximations to the kernel PDEs. Furthermore, we demonstrate that the DeepONet approximation gain kernels ensure stability when replacing the exact backstepping gain kernels. Notably, DeepONet operator exhibits computation speeds two orders of magnitude faster than PDE solvers for such gain functions, and their theoretically proven stabilizing capability is validated through simulations.

Application of Slip Interface Conditions in the Elastic Parabolic Equation

The parabolic equation method is the most effective approach for solving wave propagation problems in environments with strong vertical variations and gradual horizontal variations. This approach efficiently provides accurate solutions for problems involving sloping fluid-fluid interfaces, sloping solid-solid interfaces, and sloping solid boundaries. There remains a need for improvement for the case of sloping fluid-solid interfaces. The most promising approach to date is the combination of a single-scattering approximation and the treatment of part of the fluid layer as a solid layer with low shear speed. This is a singular perturbation, which results in rapid oscillations near the interface between the low-speed layer and the true solid layer. It is demonstrated here that the rapid oscillations may be eliminated by using slip conditions at the interface between the low-speed layer and the true solid layer. Stability problems were encountered when attempting to directly implement the slip conditions, but this problem was resolved by using an equivalent condition involving the normal derivative of the tangential displacement. This progress does not fully resolve the case of the sloping fluid-solid interface. There remains a need to find a vertical interface condition that is compatible with the slip conditions and that provides accurate results for problems involving sloping fluid-solid interfaces.

On the Nodal Set of Solutions to a Class of Nonlocal Parabolic Equations

We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: { ∂ t u ¯ − y − a ∇ ⋅ ( y a ∇ u ¯ ) = 0 a m p ; in B 1 + × ( − 1 , 0 ) − ∂ y a u ¯ = q ( x , t ) u a m p ; on B 1 × { 0 } × ( − 1 , 0 ) , \begin{equation*} \begin {cases} \partial _t \overline {u} - y^{-a} \nabla \cdot (y^a \nabla \overline {u}) = 0 \quad &\text { in } \mathbb {B}_1^+ \times (-1,0) \\ -\partial _y^a \overline {u} = q(x,t)u \quad &\text { on } B_1 \times \{0\} \times (-1,0), \end{cases} \end{equation*} where a ∈ ( − 1 , 1 ) a\in (-1,1) is a fixed parameter, B 1 + ⊂ R N + 1 \mathbb {B}_1^+\subset \mathbb {R}^{N+1} is the upper unit half ball and B 1 B_1 is the unit ball in R N \mathbb {R}^N . Our main motivation comes from its relation with a class of nonlocal parabolic equations involving the fractional power of the heat operator H s u ( x , t ) = 1 | Γ ( − s ) | ∫ − ∞ t ∫ R N [ u ( x , t ) − u ( z , τ ) ] G N ( x − z , t − τ ) ( t − τ ) 1 + s d z d τ . \begin{equation*} H^su(x,t) = \frac {1}{|\Gamma (-s)|} \int _{-\infty }^t \int _{\mathbb {R}^N} \left [u(x,t) - u(z,\tau )\right ] \frac {G_N(x-z,t-\tau )}{(t-\tau )^{1+s}} dzd\tau . \end{equation*} We characterise the possible blow-ups and we examine the structure of the nodal set of solutions vanishing with a finite order. More precisely, we prove that the nodal set has at least parabolic Hausdorff codimension one in R N × R \mathbb {R}^N\times \mathbb {R} , and can be written as the union of a locally smooth part and a singular part, which turns out to possess remarkable stratification properties. Moreover, the asymptotic behaviour of general solutions near their nodal points is classified in terms of a class of explicit polynomials of Hermite and Laguerre type, obtained as eigenfunctions to an Ornstein-Uhlenbeck type operator. Our main results are obtained through a fine blow-up analysis which relies on the monotonicity of an Almgren-Poon type quotient and some new Liouville type results for parabolic equations, combined with more classical results including Federer’s reduction principle and the parabolic Whitney’s extension.

Investigation of Fractional Order Tumor Cell Concentration Equation Using Finite Difference Method

المعادلة التفاضلية الكسرية توفر إطارًا رياضيًا قويًا لنمذجة وفهم مجموعة واسعة من الظواهر المعقدة وغير المحلية والمعتمدة على الذاكرة في مختلف التطبيقات العلمية والهندسية والعالمية. تعد الاعتلالات الكبدية أورامًا سرطانية ثانوية تتطور في الكبد نتيجة انتشار خلايا السرطان منورام سرطاني أصلي ينشأ في جزء آخر من جسم الإنسان. التركيز الأساسي لهذه الدراسة هو فهم نمو الأورام في الكبد البشري، سواء بوجود أو بدون علاج دوائي. ولتحقيق ذلك، يتم استخدام معادلة تفاضلية جزئية معرفة بالترتيب الزمني والكسري، ويتم تحليلها باستخدام أساليب عددية. يتم استخدام المشتقة كابوتو لاستكشاف تأثير العلاج الدوائي على نمو الورم. لحل النموذج الرياضي عدديًا، تم تطوير طريقة الفروق المحدودة كرانك-نيكولسون. يتم اختيار هذه الطريقة بسبب خصائصها المميزة، بما في ذلك الاستقرار الغير مشروط والدقة من الدرجة الثانية في الأبعاد الزمنية والمكانية. الاستقرار والدقة التي توفرهما هاتان الخصائص أمران حاسمان لضمان موثوقية النتائج المحصلة في هذه البحث. يتم تقديم النتائج والتحققات المستمدة من هذه الدراسة بفعالية من خلال تمثيلات بصرية متنوعة. تعتبر هذه الوسائل البصرية أدوات لا غنى عنها لفهم التأثير العميق للعلاج الدوائي على نمو الأورام. من خلال هذا التمثيل البصري، يمكن للشخص الحصول على فهم أوضح وأكثر تفصيلًا للديناميات المعقدة التي تحدث. تم الوصول إلى الحل العددي لهذه المشكلة المعقدة من خلال تنفيذ خوارزميات تم تطويرها بدقة باستخدام لغة البرمجة بايثون المتعددة الاستخدامات والقوية. مرونة بايثون والمكتبات الشاملة وإمكانيات الحساب العددي القوية تجعلها الخيار المثالي للتعامل مع تعقيدات هذه الدراسة.

Quenching behavior and critical speed for a parabolic problem due to a moving nonlinear source

This article investigates an initial–boundary value problem on the semi-infinite interval for a parabolic equation with a moving nonlinear source. The study presents criteria for both finite-time quenching and global existence of the solution. It is shown that there exists a critical speed v∗ for the nonlinear source, ensuring global existence of the solution when the speed of the moving nonlinear source is greater than or equal to v∗, while finite-time quenching occurs when the speed is smaller than v∗. The formula to calculate the critical speed v∗ is also provided for a special case.

On the modulus of continuity of solutions to nonlocal parabolic equations

AbstractA general modulus of continuity is quantified for locally bounded, local, weak solutions to nonlocal parabolic equations, under a minimal tail condition. Hölder modulus of continuity is then deduced under a slightly stronger tail condition. These regularity estimates are demonstrated under the framework of nonlocal ‐Laplacian with measurable kernels.