Efficient approximate analytical methods for nonlinear fuzzy boundary value problem

This paper aims to solve the nonlinear two-point fuzzy boundary value problem (TPFBVP) using approximate analytical methods. Most fuzzy boundary value problems cannot be solved exactly or analytically. Even if the analytical solutions exist, they may be challenging to evaluate. Therefore, approximate analytical methods may be necessary to consider the solution. Hence, there is a need to formulate new, efficient, more accurate techniques. This is the focus of this study: two approximate analytical methods-homotopy perturbation method (HPM) and the variational iteration method (VIM) is proposed. Fuzzy set theory properties are presented to formulate these methods from crisp domain to fuzzy domain to find approximate solutions of nonlinear TPFBVP. The presented algorithms can express the solution as a convergent series form. A numerical comparison of the mean errors is made between the HPM and VIM. The results show that these methods are reliable and robust. However, the comparison reveals that VIM convergence is quicker and offers a swifter approach over HPM. Hence, VIM is considered a more efficient approach for nonlinear TPFBVPs.

Time Fractional Generalized Korteweg-de Vries Equation: Explicit Series Solutions and Exact Solutions

In this article, an attempt is made to achieve the series solution of the time fractional generalized Korteweg-de Vries equation which leads to a conditionally convergent series solution. We have also resorted to another technique involving conversion of the given fractional partial differential equations to ordinary differential equations by using fractional complex transform. This technique is discussed separately for modified Riemann-Liouville and conformable derivatives. Convergence analysis and graphical view of the obtained solution are demonstrated in this work.

Probabilistic Combinatorial Model of the Epidemic for a University

The aim of the research was to identify the main causes of infection of teachers and students in a university. Two probabilistic combinatorial problems are considered analytically to determine the probabilities and rates of infection of teachers and students in a university as a result of the appearance of infected persons among the contingent of students. The mathematical apparatus of probability theory and combinatorics is used to solve the problems. For the factorials of combinations arising in the structure, the asymptotic Stirling’s formula is used. Convergent series arise in the final formulas, reflecting the multiplicity of scenarios of the probabilistic approach. Analytical formulas for the sums of series, probabilities and rates of infection of teachers and students are obtained. It is shown that the infection of teachers and students occurs through «dangerous» spatially close contacts, when a teacher and a student talk at a distance of less than 0.5 meter. It is impossible to exclude such contacts in the students’ environment during full-time study. Among teachers, there is also a less probable classroom mechanism of infection through the volume of air infected with viruses.

Banach Actions Preserving Unconditional Convergence

Let A,X,Y be Banach spaces and A×X→Y, (a,x)↦ax be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if for every bounded sequence (an)n∈ω in A and unconditionally convergent series ∑n∈ωxn in X, the series ∑n∈ωanxn is unconditionally convergent in Y. We prove that a Banach action A×X→Y preserves unconditional convergence if and only if for any linear functional y*∈Y* the operator Dy*:X→A*, Dy*(x)(a)=y*(ax) is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from ℓ1 to ℓ2, we prove that a Banach action A×X→Y preserves unconditional convergence if A is a Hilbert space possessing an orthonormal basis (en)n∈ω such that for every x∈X, the series ∑n∈ωenx is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers p,q,r∈[1,∞] with 1r≤1p+1q, the coordinatewise multiplication ℓp×ℓq→ℓr preserves unconditional convergence if and only if one of the following conditions holds: (i) p≤2 and q≤r, (ii) 2<p<q≤r, (iii) 2<p=q<r, (iv) r=∞, (v) 2≤q<p≤r, (vi) q<2<p and 1p+1q≥1r+12.

Solutions of General Fractional-Order Differential Equations by Using the Spectral Tau Method

Here, in this article, we investigate the solution of a general family of fractional-order differential equations by using the spectral Tau method in the sense of Liouville–Caputo type fractional derivatives with a linear functional argument. We use the Chebyshev polynomials of the second kind to develop a recurrence relation subjected to a certain initial condition. The behavior of the approximate series solutions are tabulated and plotted at different values of the fractional orders ν and α. The method provides an efficient convergent series solution form with easily computable coefficients. The obtained results show that the method is remarkably effective and convenient in finding solutions of fractional-order differential equations.

A Reliable Treatment for Nonlinear Differential Equations

In this paper, we use the concept of homotopy, Laplace transform, and He’s polynomials, to propose the auxiliary Laplace homotopy parameter method (ALHPM). We construct a homotopy equation consisting on two auxiliary parameters for solving nonlinear differential equations, which switch nonlinear terms with He’s polynomials. The existence of two auxiliary parameters in the homotopy equation allows us to guarantee the convergence of the obtained series. Compared with numerical techniques, the method solves nonlinear problems without any discretization and is capable to reduce computational work. We use the method for different types of singular Emden–Fowler equations. The solutions, constructed in the form of a convergent series, are in excellent agreement with the existing solutions.

Analysis of thermally stratified radiative flow of Sutterby fluid with mixed convection

In this attempt, we investigate the mixed convection in Sutterby fluid flow based on boundary layer approximation. Heat transport analysis is composed of stratification and thermal radiative phenomena. The system of non-linear PDEs is transformed into coupled ODEs. Convergent series approximations are evaluated via homotopic technique. Influence of various pertinent parameters is sketched and graphically analyzed. It is found that horizontal velocity increments for higher mixed convection parameter. The radiation parameter has a similar relation with temperature whereas the stratification parameter shows opposite behavior for temperature field.

Generalized and Multiple-Queries-Oriented Privacy Budget Strategies in Differential Privacy via Convergent Series

For data analysis with differential privacy, an analysis task usually requires multiple queries to complete, and the total budget needs to be divided into different parts and allocated to each query. However, at present, the budget allocation in differential privacy lacks efficient and general allocation strategies, and most of the research tends to adopt an average or exclusive allocation method. In this paper, we propose two series strategies for budget allocation: the geometric series and the Taylor series. We show the different characteristics of the two series and provide a calculation method for selecting the key parameters. To better reflect a user’s preference of noise during the allocation, we explored the relationship between sensitivity and noise in detail, and, based on this, we propose an optimization for the series strategies. Finally, to prevent collusion attacks and improve security, we provide three ideas for protecting the budget sequence. Both the theoretical analysis and experimental results show that our methods can support more queries and achieve higher utility. This shows that our series allocation strategies have a high degree of flexibility which can meet the user’s need and allow them to be better applied to differentially private algorithms to achieve high performance while maintaining the security.