Mild Solutions Research Articles

On nonlinear Sobolev equation with the Caputo fractional operator and exponential nonlinearity

The initial value problem for the Caputo type time‐fractional Sobolev equation with a nonlinear exponential source function is investigated in this work. We establish the existence and uniqueness of mild solutions corresponding to two different initial data assumptions. We derive global results of a unique mild solution with small initial data using some Sobolev/Sobolev‐Orlicz embeddings, a weighted Banach space, and the fixed point theorem. In the absence of any smallness assumptions, the Cauchy iteration method demonstrates that the mild solution blows up at a finite time or exists globally in time. Finally, we consider some illustrated examples to test the results obtained in theory.

Selective recovery of lithium from high-aluminum fly ash by using alkali-dissolution-assisted HBTA–TOPO synergistic extraction

The process of recycling lithium (Li) from common coal fly ash using strong acids can release substantial quantities of hazardous low-value residues, and the complex ionic composition of the acidic leaching solution hinders the selective extraction of Li. Accordingly, a green and economical approach was developed for the selective recovery of Li and extraction of valuable Al-rich residues from high-alumina fly ash (HAFA). In HAFA, the glass phase (the main Li-bearing phase) can be selectively dissolved in the mild NaOH solution at 95 °C for 2 h, and valuable mullite phase (the main Al-bearing phase) was retained in the leaching residue (Al2O3 content 49.00 wt%). The leaching efficiencies of Li, Si, and Al were 91.60%, 43.19%, and 3.56%, respectively. Among the primary metallic elements present in HAFA, only Li, Na, K, and Al in the glass phase were readily dissolved in the alkaline solution. Meanwhile, the synergistic extraction system of HBTA and TOPO has good complexing ability with Li + compared to Al3+, K+, and Na+. Therefore, Li+ can be selectively extracted from alkaline leaching solutions using HBTA and TOPO. The Li extraction efficiency reached 99.68% after three consecutive extractions under the optimal conditions. Finally, the used alkali leaching solution and synergistic extractants can be regenerated via Ca(OH)2 precipitation and HCl reverse extraction. These findings can provide new strategies for the green and overall utilization of HAFA.

Characteristics and electrochemical membrane sensors for selective study and determination of some antidepressant drugs in their pharmaceutical preparations

The construction and electrochemical characterisation of potentiometric sensors of the PVC-membrane type are described being developed for the determination of selected antidepressants; namely: Olanzapine, Oxazepam, and Lorazepam. The sensing PVC-membranes incorporate the ion-associates of the drug cations with ammonium reineckate as selective materials dispersed in 2-nitrophenyl-octylether (NPOE) or dibutyl sebacate (DBS) as the plasticizers of choice. These ISEs exhibit rapid, stable and nearly Nernstian response over a relatively wide concentration range of 1 ´ 10-7 M – 0.01 mol L-1 for Olanzapine and 1 ´ 10-6 – 0.01 mol L-1 for Oxazepam and Lorazepam, in mild acidic solutions with pH 4.5. No interferences caused by either inorganic or organic species were found. The three electrodes developed have been tested for potentiometric titrations of all three drugs, as well as for their direct determination via the respective calibration curves. The individual experiments then resulted in average recovery rates in an interval of 98.4-99.2 %, with the R.S.D. of ca. ±1.5 % (n = 3) for all three analytes of interest. Finally, the method proposed has been applied to the determination of the three drugs in pharmaceutical formulations and in urine, when the results obtained were comparable to those obtained with the reference HPLC measurements.

Response of an Euler–Bernoulli beam subject to a stochastic disturbance

This work concerns the stochastic analysis of the bending of a slender cantilever beam subject to an external force with the inclusion of a stochastic effect characterised by white noise. The beam deflection is governed by the classic dynamic Euler–Bernoulli equation. Its response to the stochastic external load is investigated by learning pattern from the simulation data which are collected from numerical computations of ten thousand numerical experiments, which are achieved using a finite difference method coupled with a Monte Carlo method for the uncertainty quantification. Insightful results are presented with visualisation techniques and discussed in detail. Of note, by performing regression analysis to the data, the solution is shown to follow a centred Gaussian process with a strong numerical evidence. The associated autocovariance matrix is computed using the sample data. Then, a mild solution in the probability sense for the deflection at a fixed position and a fixed time is written explicitly in a simple form. The results obtained by the finite difference scheme were also compared to the finite-element scheme and were found to be in good agreement. Unsurprisingly, the finite-element scheme was found to be much more computationally expensive compared to finite difference scheme. Hence, for such a simple structure, the stochastic analysis using the finite difference scheme is preferred. Analysis of the results also showed that some of the regression parameters converge when the number of simulations reaches five hundred and only vary subject to numerical errors of order 10-6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-6}$$\\end{document} if the number of simulations is further increased. While others converge when the number of simulations exceeds two thousand showing that depending on the level of precision required fewer than ten thousand simulations might be required.

Existence and controllability of non-local fractional dynamical systems with almost sectorial operators

In this article, we investigate the existence of the mild solutions and controllability of nonlocal fractional dynamical system with almost sectorial operator. The dynamical system under consideration involves Caputo fractional derivative of order α∈(0,1). The existence results are proved by using fixed point theorems with suitable assumptions. Sufficient conditions for controllability are derived using appropriate control function via Leray-Schauder fixed point theorem. An example is provided to validate the results.

Probabilistic global well-posedness to the nonlocal Degasperis–Procesi equation

We study the almost sure existence of a global solution of the nonlocal Degasperis–Procesi equation. After a suitable randomization, we obtain the local existence and uniqueness of the mild solution for a large set of the supercritical initial data. Then, we establish an energy estimate of the solution and get the global existence in probability. The methods can be used to get the well-posedness of fractional Burgers equation with the supercritical initial data.

Well-posedness results for a new class of stochastic spatio-temporal SIR-type models driven by proportional pure-jump Lévy noise

This paper provides a first attempt to incorporate the massive discontinuous changes in the spatio-temporal dynamics of epidemics. Namely, we propose an extended class of epidemic models, governed by coupled stochastic semilinear partial differential equations, driven by pure-jump Lévy noise. Based on the considered type of incidence functions, by virtue of semigroup theory, a truncation technique and Banach fixed point theorem, we prove the existence and pathwise uniqueness of mild solutions, depending continuously on the initial datum. Moreover, by means of a regularization technique, based on the resolvent operator, we acquire that mild solutions can be approximated by a suitable converging sequence of strong solutions. With this result at hand, for positive initial states, we derive the almost-sure positiveness of the obtained solutions. Finally, we present the outcome of several numerical simulations, in order to exhibit the effect of the considered type of stochastic noise, in comparison to Gaussian noise, which has been used in the previous literature. Our established results lay the ground-work for investigating other problems associated with the new proposed class of epidemic models, such as asymptotic behavior analyses, optimal control as well as identification problems, which primarily rely on the existence and uniqueness of biologically feasible solutions.

Existence results for some stochastic functional integrodifferential systems driven by Rosenblatt process

Abstract This work investigates the existence and uniqueness of mild solutions to a class of stochastic integral differential equations with various time delay driven by the Rosenblatt process. We can obtain alternative conditions that guarantee mild solutions by using the resolvent operator in the Grimmer sense, stochastic analysis, fixed-point methods, and noncompact measures. We give an example to illustrate the theory.

On the global existence and analyticity of the mild solution for the fractional Porous medium equation

In this research article we focus on the study of existence of global solution for a three-dimensional fractional Porous medium equation. The main objectives of studying the fractional porous medium equation in the corresponding critical function spaces are to show the existence of unique global mild solution under the condition of small initial data. Applying Fourier transform methods gives an equivalent integral equation of the model equation. The linear and nonlinear terms are then estimated in the corresponding Lei and Lin spaces. Further, the analyticity of solution to the fractional Porous medium equation is also obtained.

Existence of mild solutions for a coupled system of fractional evolution equations via Mönch's fixed point theorem in Banach spaces

Existence of mild solutions for a coupled system of fractional evolution equations via Mönch's fixed point theorem in Banach spaces

Monotone iterative technique for evolution equations with delay and nonlocal conditions in ordered Banach space

In this paper, based on monotone iterative method in the presence of the lower and upper solutions, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of nonlocal problems of evolution equations with delay in ordered Banach spaces. Firstly, we introduce the concept of lower S-asymptotically ω-periodic solution and upper S-asymptotically ω-periodic solution. Secondly, we construct monotone iterative method in the presence of the lower and upper solutions to evolution equations with delay, and obtain the existence of maximal and minimal S-asymptotically ω-periodic mild solutions for the mentioned system under wide monotone conditions and noncompactness measure condition of nonlinear term. Finally, as the application of abstract results, an example is given to illustrate our main results.

Stochastic Quasi-Geostrophic Equation with Jump Noise in Lp Spaces

In this paper, we consider a 2D stochastic quasi-geostrophic equation driven by jump noise in a smooth bounded domain. We prove the local existence and uniqueness of mild Lp(D)-solutions for the dissipative quasi-geostrophic equation with a full range of subcritical powers α∈(12,1] by using the semigroup theory and fixed point theorem. Our approach, based on the Yosida approximation argument and Itô formula for the Banach space valued processes, allows for establishing some uniform bounds for the mild solutions and we prove the global existence of mild solutions in L∞(0,T;Lp(D)) space for all p>22α−1, which is consistent with the deterministic case.

Approximation of solutions to integro-differential time fractional order parabolic equations in L^{p}-spaces

In this paper we study the initial boundary value problem for a class of integro-differential time fractional order parabolic equations with a small positive parameter ε. Using the Laplace transform, Mittag-Leffler operator family, C_{0}-semigroup, resolvent operator, and weighted function space, we get the existence of a mild solution. For suitable indices pin [1,+infty ) and sin (1,+infty ), we first prove that the mild solution of the approximating problem converges to that of the corresponding limit problem in L^{p}((0,T), L^{s}(Omega )) as varepsilon rightarrow 0^{+}. Then for the linear approximating problem with ε and the corresponding limit problem, we give the continuous dependence of the solutions. Finally, for a class of semilinear approximating problems and the corresponding limit problems with initial data in L^{s}(Omega ), we prove the local existence and uniqueness of the mild solution and then give the continuous dependence on the initial data.

Experimenting with Dimethyl Sulfoxide to Leach Gold from a Colombian Artisanal Gold Ore

The diverse uses of gold and its crucial role in the global economy are growing, particularly during cycles of economic crises. The broad use of cyanide by conventional gold-mining companies and mercury by artisanal miners poses environmental and health concerns for local communities. This article introduces an innovative gold-leaching process using a non-toxic organic reagent, dimethyl sulfoxide (DMSO), a water-free lixiviant that extracts gold from ores/concentrates in combination with copper halides. The results of laboratory experiments using dimethyl sulfoxide and a sample of high-grade gold ore from Colombia show that 96.5% of the gold was extracted in 2 h at room temperature. The typical cyanidation process using 5 g/L of CN− at pH 10.5 on the same ore sample obtained 97% gold extraction in 24 h at ambient temperature. The gold extracted using DMSO was precipitated by adding a mild acidic solution, and the reagent can be recycled via distillation and reused in repeating cycles. The results show that DMSO can be used as a promising agent for gold leaching, offering a straightforward, cost-effective, and eco-friendly procedure with minimal chemical waste.

Long-time dynamics of classical Keller–Segel equation

In this article, we obtain global dynamics of 2D Keller–Segel equation (KS for short) in subcritical regime with no additional assumptions. With initial data small in some Triebel-Lizorkin space, we also characterize the long-time asymptotics of global mild solutions to the KS equation in higher dimensions where the spatial dimension n≥3.

Asymptotically almost periodic mild solutions for some partial integrodifferential inclusions using scale of Banach spaces

Abstract We are interested in the existence of mild solutions for a class of partial integrodifferential inclusions in infinite dimensional Banach spaces. First, we show the existence of mild solutions with the help of a scale of Banach spaces, the theory of resolvent operators, and the fixed point theory for the measure of non-compactness. Moreover, we examine the existence of asymptotically almost periodic solutions for our problem. Finally, an example of the abstract results is provided.

Stochastic McKean–Vlasov equations with Lévy noise: Existence, attractiveness and stability

This work considers McKean–Vlasov equations driven by Lévy noise in which the coefficients rely not only on the state of the unknown process but also on its probability distribution, and researches the well-posed, attractiveness and stability of solutions with resolvent operator theory. Under global Lipschitz condition, the well-posed of mild solutions is first studied for McKean–Vlasov integro-differential equations by using contraction mapping principle. Then according to Cauchy–Schwartz inequality and Itô isometry formula, one gains global attracting set and quasi-invariant set of solutions of McKean–Vlasov integro-differential equations and also obtained sufficient conditions of stability with continuous dependence on the coefficient of solutions. Moreover, one attains mean-square stability of solutions by Gronwall inequality and almost sure stability by applying Borel–Cantelli lemma and Chebyshev inequality for the aforementioned equation. Finally two examples are presented to verify our results.

On initial value problem for elliptic equation on the plane under Caputo derivative

Abstract In this article, we are interested to study the elliptic equation under the Caputo derivative. We obtain several regularity results for the mild solution based on various assumptions of the input data. In addition, we derive the lower bound of the mild solution in the appropriate space. The main tool of the analysis estimation for the mild solution is based on the bound of the Mittag-Leffler functions, combined with analysis in Hilbert scales space. Moreover, we provide a regularized solution for our problem using the Fourier truncation method. We also obtain the error estimate between the regularized solution and the mild solution. Our current article seems to be the first study to deal with elliptic equations with Caputo derivatives on the unbounded domain.

Global existence of solutions for Boussinesq system with energy dissipation

The Boussinesq system coupled by an energy dissipation brings new challenges in the study of global existence of solutions, for instance, this system does not have scale invariance which makes it difficult to show existence of mild solutions when initial data belongs to scaling invariant function spaces. In this paper we are interested to show global existence of solutions [u,θ] for Boussinesq system coupled by bilinear energy dissipation Φ(u)=2μE(u)⋅E(u) on smooth bounded domain Ω⊆Rn or whole space Rn, n⩾3, when the initial data [u0,θ0]∈X0=Wσ1,n/2(Ω)×Ln/2(Ω) is sufficiently small and the external force F(θ)=ϱfθen has low regularity in the sense t1/2−b/2nf∈L∞((0,T):Lb(Ω)) or f∈Ls((0,T):Lb(Ω)), where 2s=1−nb and b∈[n,∞). It seems that our results on the global and local well-posedness are the first to provide an Lp-approach when the initial velocity data u0 belongs to the space W˙σ1,n/2(Rn) that is scaling invariant.

Approximate controllability of higher order Riemann–Liouville fractional systems via cosine family

This work studies the controllability of a class of Riemann–Liouville fractional differential equations of order in a new Banach space using fractional cosine family. To derive the existence of solutions, we construct the definition of mild solutions of the system using Laplace transform theory. Then, by assuming the approximate controllability of corresponding linear equation, we prove that the nonlinear dynamical system is approximately controllable using an iterative technique. Lastly, we give an example to illustrate the established theory.