Nonhomogeneous Case Research Articles

A novel numerical approach for the third order Emden–Fowler type equations

AbstractThis article aims to achieve robust numerical results by applying the Chebyshev reproducing kernel method without homogenizing the initial‐boundary conditions of the Emden–Fowler (E‐F) equation, thereby introducing a new perspective to the literature. A novel numerical approach is presented for solving the initial‐boundary value problem of third‐order E‐F equations using Chebyshev reproducing kernel theory. Unlike previous applications, which were confined to homogeneous initial‐boundary value problems or required homogenization, the proposed method is effective for both homogeneous and nonhomogeneous cases. To handle the initial‐boundary conditions of the E‐F equations, additional basis functions are introduced rather than imposing conditions on the reproducing kernel Hilbert space. The method's effectiveness is demonstrated through five examples, which validate the theoretical analysis. Overall, the results emphasize the method's efficiency.

Non-homogeneous model for studying femur bone implants with considering defects

ABSTRACT In this article, the natural frequencies of the femur in the presence and absence of defects have been calculated. For this purpose, two approaches to bone modeling have been considered: one for the homogeneous case and the other for the non-homogeneous case, which more closely reflects the reality of the femur bone. In the next step, in order to identify suitable materials for femur bone implants, two materials AZ31 and CP Ti have been selected, and the natural frequencies of the femur made with these two materials have been calculated and compared with the results of the natural femur. From the obtained results, it can be concluded that although these two materials are attractive for bone implants from other aspects, they are not approved from the point of view of vibrations, although material AZ31 is a little better. It is also observed that the selected defect in the femur has little effect on the natural frequencies.

Fuzzy Langevin fractional delay differential equations under granular derivative

Analytical studies of the class of the fuzzy Langevin fractional delay differential equations (FLFDDEs) are frequently complex and challenging. It is necessary to construct an effective technique for the solution of FLFDDEs. This article presents an explicit analytical representation of the solution to the class of FLFDDEs with the general fractional orders under granular differentiability. The closed-form solution to the FLFDDEs is extracted for both the homogeneous and non-homogeneous cases using the Laplace transform technique and presented in terms of the delayed Mittag-Leffler type function with double infinite series. Moreover, the existence and uniqueness of the solutions of the FLFDDEs are investigated using the generalized contraction principle. An illustrative example is provided to support the proposed technique. To add to the originality of the presented work, the FLFDDEs with constant delay are solved by applying vibration theory and visualizing their graphs to support the theoretical results.

Three Regularization Methods for A Homogeneous Time Fractional Diffusion Equation with Space - Dependent Diffusivity Coefficient

We studied an inverse problem for a time fractional diffusion equation with space - dependent diffusivity coefficient in the nonhomogeneous case. This problem is ill – posed in Hadamard’s sense. So, a regularization is essential. Using three regularization methods, we get the estimation of the error between the regularized solution and the exact solution

An innovative iterative approach to solving Volterra integral equations of second kind

Many scientists have shown great interest in exploring the realm of second-kind integral equations, offering many techniques for solving them, including exact, approximate, and numerical methods. This paper introduces the Hussein-Jassim method (HJ-method) for solving Volterra integral equations of the second kind (VIESKs). The foundation of this approach lies in the principle of Maclaurin expansion. The algorithm of the method was derived, and its convergence was analysed. Furthermore, the method was applied to various Volterra integral equations, encompassing linear, nonlinear, homogeneous, and nonhomogeneous cases. Ultimately, the proposed method successfully addressed these equations, with the approximate solutions converging toward the exact solution.

Numerical modeling of temperature-reporting nanoparticle tracer for fractured geothermal reservoir characterization

Information on the temperature distribution of subsurface reservoirs is essential for geothermal energy development. One of the promising tools to detect the reservoir temperature distribution is temperature-reporting nanoparticle tracers whose functionality has been extensively investigated in both theoretical and experimental ways in the last decade. However, most related studies were limited to simplified geometries and ignored the dynamic interplays of fluid flow, heat transfer, transport and reaction of the temperature-reporting nanoparticle tracer. The response behavior and working mechanisms of such nanotracers in a realistic three-dimensional system still have not been fully revealed through a systematic study. In this work, we develop a numerical modeling approach to simulate field implementation of these nanotracers in a fractured geothermal reservoir. This study aims to evaluate whether the injection of multiple temperature-reporting nanoparticle tracers with different thresholds can be used to estimate the temperature distribution and provide information on the thermal and geological heterogeneities. Several scenarios have been investigated for the geothermal reservoir including homogeneous and non-homogeneous cases (e.g., thermal and geological heterogeneities). Our obtained results from the nanotracer breakthrough curves show that the deviation temperatures in peak concentration values provide an upper limit of the lowest temperature and precise highest temperature for the reservoir temperature range. The deviation temperature of the peak arrival time curve accurately estimates the highest temperature along the main streamlines between the wells. The proposed analysis curves based on the nanotracer breakthrough data were visibly affected by geological heterogeneities including their conductivities and orientations as well as thermal heterogeneities in the geothermal reservoir.

Recovering the Heat Distribution on the Surface of a Two-dimensional Finite Slab from Interior Data in the Nonhomogeneous Case

We considered the two – dimensional problem of reconstructing the historical distribution on the surface of a finite slab from interior temperature data in the nonhomogeneous case. The problem is ill – posed. So, a regularization is essential. Using the integration truncation method, we have got the estimation of the error between the regularized solution and the exact solution in the nonhomogeneous case. Then, we provided a numerical experiment for illustration ofthe theoretically obtained results.

The Time-Dependent Symbol of a Non-homogeneous Itô Process and Corresponding Maximal Inequalities

The probabilistic symbol is defined as the right-hand side derivative at time zero of the characteristic functions corresponding to the one-dimensional marginals of a time-homogeneous stochastic process. As described in various contributions to this topic, the symbol contains crucial information concerning the process. When leaving time-homogeneity behind, a modification of the symbol by inserting a time component is needed. In the present article, we show the existence of such a time-dependent symbol for non-homogeneous Itô processes. Moreover, for this class of processes, we derive maximal inequalities which we apply to generalize the Blumenthal–Getoor indices to the non-homogeneous case. These are utilized to derive several properties regarding the paths of the process, including the asymptotic behavior of the sample paths, the existence of exponential moments and the finiteness of p-variationa. In contrast to many situations where non-homogeneous Markov processes are involved, the space-time process cannot be utilized when considering maximal inequalities.

On the weighted inertia-energy approach to forced wave equations

We show convergence of minimizers of weighted inertia-energy functionals to solutions of initial value problems for a class of nonlinear wave equations. The result is given for the nonhomogeneous case under a natural growth assumption on the forcing term.

Explicit analytical solutions of an incommensurate system of fractional differential equations in a fuzzy environment

Fuzzy fractional models have attracted considerable attention because of their comprehensive and broader understanding of real-world problems. Analytical studies of these models are often complex and difficult. Therefore, it is beneficial to develop a suitable and comprehensive technique to solve these models analytically. In this paper, an explicit analytical technique for solving two-dimensional incommensurate linear fuzzy systems of fractional Caputo differential equations (FLSoCFDEs) considering generalized Hukuhara differentiability (gH-differentiability) is presented and demonstrated. This extracted explicit solution is presented for different classes of such systems, including homogeneous and non-homogeneous cases with commensurate and incommensurate fractional orders. Moreover, the potential solution of FLSoCFDEs in terms of the Mittag-Leffler function involving double series is presented. The originality of the proposed technique is that the fuzzy Cauchy problem is transformed into a system of fuzzy linear Volterra integral equations of second kind and then the solution is extracted using the iterative Picard scheme based on the Banach fixed point theorem. Moreover, several interesting results are derived from FLSoCFDEs in terms of the Mittage-Leffler function for both homogeneous and inhomogeneous cases. To understand the proposed technique, we solve a diffusion process problem (a biological model) and several mass-spring systems as applications. Their graphs are analyzed to illustrate and support the theoretical results.

Efficient 2D Tikhonov smoothness regularization with recursive filtering

In this work, we describe an implementation of the 2D Tikhonov regularization filter which scales linearly with the input signal’s size. In the homogeneous case, we propose a novel algorithm to decompose the filter’s 2D kernel as a sum of axis-aligned Gaussians. Our algorithm uses symmetries of the kernel to provide a fast computation of the Gaussian decomposition in the frequency domain, where the 2D Tikhonov kernel has a closed-form expression. The convolution with each Gaussian is then computed using linear-time separable recursive filtering. In the non-homogeneous case, we also decompose the 2D problem as a series of iterated linear-time separable recursive filters, which can be combined with the Biconjugate Gradient Stabilized method for fast convergence. In this way, a fast solution to the 2D Tikhonov regularization problem is obtained.

Systems of Linear Equations with Non-Negativity Constraints: Hyper-Rectangle Cover Theory and Its Applications

In this paper, a novel hyper-rectangle cover theory is developed. Two important concepts, the cover order and the cover length, are introduced. We construct a specific échelon form of the matrix in the same manner as that employed to determine the rank of the matrix to obtain the cover order of any given matrix. Using the properties of the cover order, we obtain the necessary and sufficient conditions for the existence and uniqueness of the solutions for linear equations system with non-negativity constraints on variables for both homogeneous and nonhomogeneous cases. In addition, we apply the cover theory to analyze some typical problems in linear algebra and optimization with non-negativity constraints on variables, including linear programming (LP) problems and non-negative least squares (NNLS) problems. For LP problems, the three possible behaviours of the solutions are studied through cover theory. On the other hand, we develop a method to obtain the cover length of the covered variable. In this process, we discover the relationship between the cover length determination problem and the NNLS problem. This enables us to obtain an analytical optimal value for the NNLS problem.

Semi-analytical solution to steady-state and transient heat transfer problem with variable conductivity properties of the domain

Introduction. Present paper studies the case of heat transfer problem where the conductivity property of the domain is not a constant value but varies depending on its temperature. In particular, a conductivity that linearly dependent onthe temperature is reviewed. A semi-analytical formulation is developed and proposed to solve steady-state and transient types of these kinds of heat transfer problems.
 Materials and methods. The domain is mathematically modeled by using finite element method where non-linearity property of the problem has been incorporated. For time-dependent (transient) case, the temporal solution has been achieved by analytical methods wherefore all necessary formulation has been derived. Non-linear equations have been solved by both Newton and Picard methods.
 Results. In order to proof check, the proposed calculations a wall with three layers has been calculated by using the proposed transient problem formulation. The derivative boundary conditions at the faces of the wall are given as the functions of time. Non-linearity has been solved by two different methods that gave the identical results. Having solved the non-linearity by these two methods, allowed to compare the efficiency of Newton method versus Picard method. Both methods reached the solution with the same number of iterations. The paper proposes the algorithms for solving the problems. The authors have used MatLab environment to implement those algorithms.
 Conclusions. Proposed formulation solves coupled heat transfer problems in multi-layered exterior walls. Multi-layered calculation ability allows to cover all non-homogeneous cases of the computation domain. The formulation solves the heat loss problems through the multi-layered walls with due and reliable accuracy.

A half-discrete Hilbert-type inequality in the whole plane with the constant factor related to a cotangent function

In this work, by the introduction of some parameters, a new half-discrete kernel function in the whole plane is defined, which involves both the homogeneous and the nonhomogeneous cases. By employing some techniques of real analysis, especially the method of a weight function, a new half-discrete Hilbert-type inequality with the new kernel function, as well as its equivalent Hardy-type inequalities are established. Moreover, it is proved that the constant factors of the newly obtained inequalities are the best possible. Finally, assigning special values to the parameters, some new half-discrete Hilbert-type inequalities with special kernels are presented at the end of the paper.

Oscillation criteria of second order differential equations with positive and negative coefficients

In this paper we obtain oscillation criteria for solutions of homogeneous and nonhomogeneous cases of second order neutral differential equations with positive and negative coefficients. Our results improve and extend the results of [Oscillation criteria for a class of second order neutral delay differential equations, Appl. Math. Comput. \textbf{210}, 303--312, 2009].

On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions

Abstract In this work, by the introduction of a new kernel function composed of exponent functions with several parameters, and using the method of weight coefficient, Hermite-Hadamard’s inequality, and some other techniques of real analysis, a more accurate half-discrete Hilbert-type inequality including both the homogeneous and non-homogeneous cases is established. Furthermore, by introducing the Bernoulli number and the rational fraction expansion of tangent function, some special and interesting Hilbert-type inequalities and their equivalent hardy-type inequalities are presented at the end of the paper.

Lagrangian evolution of field gradient tensor invariants in magneto-hydrodynamic theory

In 1982 in a series of works Vielliefosse [1, 2] discussed a nonlinear homogeneous evolution equation for the velocity gradient tensor in fluid dynamics. Later Cantwell [3] extended this formalism to the non-homogeneous case including the effects of viscous diffusion and cross derivatives of pressure field. Here, we derive the evolution equations of the geometrical invariants of the magnetic and velocity field gradient tensors in the case of magneto-hydrodynamics for both non-homogeneous and homogeneous cases, i.e., considering or neglecting viscous effects and source terms. The inclusion of dissipation effects and higher-order gradient terms introduces a non trivial evolution of invariants, which can be treated as a stochastic evolution equation. Conversely, in the homogeneous case, the magnetic field invariants do not evolve, i.e., the magnetic field line topology is conserved, while the corresponding velocity invariants are affected by magnetic forces. By writing the equations of the velocity field invariants as a dynamical system we can identify the role of the different terms in the evolution equations. In detail, in the homogenous case we show that the term associated with the current density drives transitions between hyperbolic and elliptical structures. Evolution equations are also discussed in the perspective of an application to the analysis of magneto-hydrodynamic turbulence.

The embedding property of the scaling limit of modulation spaces

Modulation spaces Mp,qs were introduced by Feichtinger [3] in 1983. Later, considering the scaling property of the modulation spaces, Sugimoto and Wang [11] defined the scaling limit of the modulation spaces, which contains both the modulation spaces and Bényi and Oh's modulation spaces [1], and these spaces also have some applications in nonlinear Schrödinger equations. So, it is important to consider the relationship between these new spaces and some classical Banach spaces such as Lp spaces, Fourier Lp spaces and Besov-Triebel-Sobolev spaces. We study the embedding properties of the scaling limit of the modulation spaces, including the homogeneous case and nonhomogeneous case.

Two-level Overlapping Schwarz Methods Based on Local Generalized Eigenproblems for Hermitian Variational Problems

The research of the two-level overlapping Schwarz (TL-OS) methods based on constrained energy minimizing coarse space is still in its infancy, and there exist some defects, e.g., mainly for a second order elliptic problem and too heavy computational cost of coarse space construction. In this paper, by introducing appropriate assumptions, we propose more concise coarse basis functions for general Hermitian positive and definite discrete systems, and establish the algorithmic and theoretical frameworks of the corresponding TL-OS methods. Furthermore, to enhance the practicability of the algorithm, we design two economical TL-OS preconditioners and prove the condition number estimate. As the first application of the frameworks, we prove that the assumptions hold for the linear finite element discretization of a second order elliptic problem with high contrast and oscillatory coefficient and the condition number of the TL-OS preconditioned system is robust with respect to the model and mesh parameters. In particular, we also prove that the condition number of the economically preconditioned system is independent of the jump range under a certain jump distribution. Experimental results show that the first kind of economical preconditioner is more efficient and stable than the existing one. Second, we construct TL-OS and the economical TL-OS preconditioners for the plane wave least squares discrete system of the Helmholtz equation by using the frameworks. The numerical results for homogeneous and nonhomogeneous cases illustrate that the preconditioned conjugate gradient method based on the proposed preconditioners has good stability in terms of the angular frequency, mesh parameters, and the number of degrees of freedom in each element.

New analysis of fuzzy fractional Langevin differential equations in Caputo's derivative sense

<abstract> <p>The extraction of analytical solution of uncertain fractional Langevin differential equations involving two independent fractional-order is frequently complex and difficult. As a result, developing a proper and comprehensive technique for the solution of this problem is very essential. In this article, we determine the explicit and analytical fuzzy solution for various classes of the fuzzy fractional Langevin differential equations (FFLDEs) with two independent fractional-orders both in homogeneous and non-homogeneous cases. The potential solution of FFLDEs is also extracted using the fuzzy Laplace transformation technique. Furthermore, the solution of FFLDEs is defined in terms of bivariate and trivariate Mittag-Leffler functions both in the general and special forms. FFLDEs are a new topic having many applications in science and engineering then to grasp the novelty of this work, we connect FFLDEs with RLC electrical circuit to visualize and support the theoretical results.</p></abstract>