Picard Method Research Articles

Identification of genetic variants by using a whole exome sequencing approach in Taiwanese patients with Brugada Syndrome

Abstract Background Brugada Syndrome (BrS) is a life-threatening arrhythmia disease primarily affecting young men. Many genetic studies have been conducted mainly in Caucasian BrS patients to identify potential genetic drivers. SCN5A mutations are less common in Asian BrS patients (around 10%) compared to Caucasian patients (20-30%). This means the pathogenic process of more than 70% of BrS patients remains unclear. With the advancement in NGS technologies, we can conduct genome-wide screening of DNA variants in BrS patients efficiently. Purpose Since less than 30% of the BrS patients have known genetic drivers for the disease, we aimed to identify more DNA variants in BrS patients by using the whole exome sequencing (WES) approach. Furthermore, we evaluated the landscape of DNA variants in the two groups of BrS patients according to whether they have SCN5A mutations or not. Methods A total of 201 BrS patients were recruited for the WES experiments. After obtaining the raw sequencing data, we performed the quality check procedures using the multiQC and Trimmomatic softwares (Figure 1). For reads passing the quality check, the BWA algorithm was used for the alignment. Next, the samtools, bamtools, and Picard methods were used to remove duplicates and perform the sorting. Subsequently, those sorted sequencing files were used to call the DNA variants by using the GATK approach and annotate them using the ANNOVAR method. The 201 BrS patients were classified into two different groups based on their mutation status of SCN5A, presentation of sudden cardiac death (SCD), and SCD along with syncope. The gene-based algorithm, SKAT, was used to analyze the summarized effects of DNA variants located in a single gene. For each DNA variant, Fisher's exact test was used to compare the allele frequency in the BrS patients under their groupings. Results The WES approach in the 201 BrS patients revealed a total of 205,454 variants, which were further categorized into two groups, common variants (51,867) and rare variants (153,587), based on their allele frequencies (5%). The SKAT algorithm was performed in the 30 genes that have been previously reported to be associated with BrS. No significant differences were observed in any gene in BrS patients for the common variants. Intriguingly, KCNJ8, SCN1B, ZFPM2, MAPRE2, and KCNE5 showed significant differences in allele frequencies between the SCN5A+ and SCN5A- patients (Figure 2). Fisher's exact test further revealed that three DNA variants (chr19:35524980, chr12:21926440, chr6:126210346) were significantly enriched in SCN5A+ patients. The 5 DNA variants located in ZFPM2 showed a linkage disequilibrium block and were enriched in SCN5A+ patients. Conclusions The WES approach in BrS patients revealed DNA variants that were reported in previous studies. Several DNA variants were significantly enriched in SCN5A+ patients, suggesting that the genome-wide mutation landscape was different in SCN5A+ and SCN5A- BrS patients.Figure 1Figure 2

Nonlinear vibrations of temperature-dependent FGM cylindrical panels subjected to instantaneous surface heating

An analysis is performed in this research to deal with the large amplitude vibrations caused by rapid surface heating in an FGM cylindrical panel. It is assumed that the properties of the panel are distributed through the thickness in terms of volume fraction of constituents. Following the first-order shear deformation theory, von Karman type of kinematics, and Donnell kinematic assumptions, definition of kinetic and strain energies of the shell are established. These obtained equations are discretized via the method of Ritz, where the shape functions are estimated by the Chebyshev polynomials. Temperature distribution within the body is also obtained using the one-dimensional heat conduction equation. This equation is discretized using the finite different method and solved iteratively following the Picard and Crank-Nicolson methods. Thermally induced force and moment resultants are evaluated and inserted into the matrix representation of equations of motion. The temporal evolution of displacement is obtained using the iterative Picard method and Newmark time marching method. Results of this study are compared with the available data in the open literature and after that novel results are given to explore the effects of geometrical, mechanical, and thermal parameters. It is highlighted that power law index, shallowness, thickness, edge supports, temperature dependency, geometrical nonlinearity and thermal boundary conditions are all important factors on the response of the structure under thermal shock. It is highlighted that thermally induced vibrations indeed exists especially in thin class of shells. Besides, thermally induced deflections may be controlled through using a proper power law index.

Picard and Adomian decomposition methods for a fractional quadratic integral equation via generalized fractional integral

The primary focus of this paper is to thoroughly examine and analyze a class of a fractional quadraticintegral equation via generalized fractional integral. To achieve this, we introduce an operator that possessesfixed points corresponding to the solutions of the fractional quadratic integral equation, effectively transforming thegiven equation into an equivalent fixed-point problem. By applying the Banach fixed-point theorems, we prove theuniqueness of solutions to fractional quadratic integral equation. Additionally, The Adomian decomposition methodis used, to solve the resulting fractional quadratic integral equation. This technique rapidly provides convergentsuccessive approximations of the exact solution to the given fractional quadratic integral equation, therefore, weinvestigate the convergence of approximate solutions, using the Adomian decomposition method. Finally, we providesome examples, to demonstrate our results. Our findings contribute to the current understanding of fractionalquadratic integral equation and their solutions and have the potential to inform future research in this area.

A handy analytical approximate solution for the magnetohydrodynamic flow of blood in a porous channel

This work presents a new version of the Picard method, known as the boundary values problems Picard method (BVPP), to obtain an analytical approximate solution for a highly complex nonlinear differential equation that models the magnetohydrodynamic flow of blood through a porous channel. The proposed method is versatile and can produce compact and easily evaluated analytical expressions that accurately capture the scientific phenomena being studied, making it ideal for practical applications. BVPP transforms a differential equation into an integral equation and utilizes an iterative algorithm like that of the basic Picard method. However, unlike the basic method, BVPP allows for the selection of an appropriate initial function and involves several adjustable parameters that can be optimized to obtain a precise analytical approximate solution with minimal effort. Overall, BVPP represents a significant advancement in the analysis of complex nonlinear differential equations, particularly in the field of biomedical engineering.

Practical Aspects for Applying Picard Iterations to the SIR Model Using Actual Data

The updated version of the Picard method for solving systems of differential equations is employed to solve the SIR system. A local performance of the Picard iteration algorithm combined with the Gauss-Seidel approach is applied to the SIR model. The integral form of the SIR model, in addition to the use of Gauss-Seidel philosophy (using the most recent calculated values), achieved more accuracy in the computational work than those obtained using the differential forms. Documented data regarding the spread of corona virus 19 in the Kingdom of Saudi Arabia region from April to the end of December 2020 were used to calculate the corresponding actual values for the parameters and the initial conditions. Due to efficient management and to obtain representable behaviours, we restricted the size of the study to only 1% of the population. The global characteristics of the integral formulation have affected the calculations accurately. The initial conditions and the model’s parameters are established depending on the documented data. The results illustrate the superiority of the updated Picard formulation over the classical Picard within their domain of convergence. The results of this study illuminate and validate the importance of mathematical modeling. These findings can provide valuable insights into mathematical modeling for those involved in environmental health research, especially those responsible for devising strategic plans.

Solutions of Time-Space Fractional Partial Differential Equations Using Picard's Iterative Method

Abstract In this paper, we present Picard's iterative method (PIM) for solving time–space fractional partial differential equations, where the derivatives are considered in the Caputo sense. We prove the existence and uniqueness of solutions. Additionally, we demonstrate the versatility of our proposed approach by obtaining exact solutions for a diverse set of equations. This method is user-friendly and directly applicable to any computer algebra system. The proposed method avoids intricate computations associated with the Adomian decomposition method, such as calculating Adomian polynomials, or the requirements of other methods like choosing a homotopy in the homotopy perturbation method, identification and manipulation of the invariant subspace in invariant subspace method or constructing a variational function in the variational iteration method. Thus, the proposed method is a versatile and efficient tool for exploring systems that involve both temporal and spatial fractional derivatives.

Picard and Adomian solutions of nonlinear fractional differential equations system containing Atangana – Baleanu derivative

In this paper, we apply two methods for solving nonlinear system of fractional differential equations (FDEs); these two methods are Picard and Adomian decomposition methods (ADM). The type of fractional derivative in this system will be the Atangana–Baleanu derivative. The existence and uniqueness of the solution will be proved. In addition, the convergence of ADM series solution and the maximum expected error will be discussed. Some numerical examples will be solved by using these two method and a comparison between their solutions will be done. There exist an important application to these types of systems, this application is the fractional-order rabies model and it will be solved here. From the obtained results, it is noticed that the obtained results from using these two methods are coincide with each other, and also these results are coincide with the obtained results from the classical fractional derivatives such as Caputo sense.

Numerical and analytical approach to the Chandrasekhar quadratic functional integral equation using Picard and Adomian decomposition methods

<p>This research aimed to find numerical solutions to a type of nonlinear initial value problem (IVP) for hybrid fractional differential equations. Using the Adomian decomposition method (ADM) and the Picard method (PM), we studied the Chandrasekhar quadratic integral equation (QIE). Furthermore, we investigated existence and uniqueness results using measures of weak noncompactness. Through a set of examples and numerical simulations, a comparison was made between the results of the AMD and PM.</p>

Fractional Integration via Picard Method for Solving Fractional Differential‐Algebraic Systems

In this paper, we applied an efficient integrative method called the fractional Picard method to find the approximate solution to a system of fractional algebraic differential equations (SFADEs). By comparing the results with the exact solution, it was found that this method is highly efficient for finding solutions. Also, a genetic algorithm (GA) was used to speed up the approximate solutions when choosing the best values for the fractional derivative, which increased the efficiency of Picard’s fractional integral method in finding the best solutions. The MATLAB program was used to find approximate solutions.

Fractional tempered differential equations depending on arbitrary kernels

<abstract><p>In this paper, we expanded the concept of tempered fractional derivatives within both the Riemann-Liouville and Caputo frameworks, introducing a novel class of fractional operators. These operators are characterized by their dependence on a specific arbitrary smooth function. We then investigated the existence and uniqueness of solutions for a particular class of fractional differential equations, subject to specified initial conditions. To aid our analysis, we introduced and demonstrated the application of Picard's iteration method. Additionally, we utilized the Gronwall inequality to explore the stability of the system under examination. Finally, we studied the attractivity of the solutions, establishing the existence of at least one attractive solution for the system. Throughout the paper, we provide examples and remarks to support and reinforce our findings.</p></abstract>

Solving the time fractional q-deformed tanh-Gordon equation: A theoretical analysis using controlled Picard's transform method

<p>This paper presented the formulation and solution of the time fractional q-deformed tanh-Gordon equation, a new extension to the traditional tanh-Gordon equation using fractional calculus, and a q-deformation parameter. This extension aimed to better model physical systems with violated symmetries. The approach taken involved the controlled Picard method combined with the Laplace transform technique and the Caputo fractional derivative to find solutions to this equation. Our results indicated that the method was effective and highlighted our approach in addressing this equation. We explored both the existence and the uniqueness of the solution, and included various 2D and 3D graphs to illustrate how different parameters affect the solution's behavior. This work aimed to contribute to the theoretical framework of mathematical physics and has potential applications across multiple interdisciplinary fields.</p>

Analysis of Inverse Coefficient Problem for Euler-Bernoulli Equation with Periodic and Integral Conditions

The research, we investigate the solution of the inverse problem of a linear Euler-Bernoulli equation. For this purpose, the existence of this problem, its uniqueness and its constant dependence on the data are demonstrated using the Picard and Fourier methods.

Complexity for a class of elliptic ordinary integro-differential equations

Consider the variational form of the ordinary integro-differential equation (OIDE)−u″+u+∫01q(⋅,y)u(y)dy=f on the unit interval I, subject to homogeneous Neumann boundary conditions. Here, f and q respectively belong to the unit ball of Hr(I) and the ball of radius M1 of Hs(I2), where M1∈[0,1). For ε>0, we want to compute ε-approximations for this problem, measuring error in the H1(I) sense in the worst case setting. Assuming that standard information is admissible, we find that the nth minimal error is Θ(n−min⁡{r,s/2}), so that the information ε-complexity is Θ(ε−1/min⁡{r,s/2}); moreover, finite element methods of degree max⁡{r,s} are minimal-error algorithms. We use a Picard method to approximate the solution of the resulting linear systems, since Gaussian elimination will be too expensive. We find that the total ε-complexity of the problem is at least Ω(ε−1/min⁡{r,s/2}) and at most O(ε−1/min⁡{r,s/2}ln⁡ε−1), the upper bound being attained by using O(ln⁡ε−1) Picard iterations.

Generalized multiscale finite element method for language competition modeling I: Offline approach

This paper develops a multiscale solver for the problem of two languages competing in a heterogeneous medium. The mathematical model contains terms for language group switching, diglossia, population growth, and diffusive-convective spatial spread. The fine grid approximation is based on the finite element method. We use the finite difference method with an implicit scheme for time approximation. We use the Picard method to linearize the nonlinear terms. We use the Generalized Multiscale Finite Element Method for approximation on a coarse grid. Numerical results are presented for a model two-language interaction problem. The results show that the proposed multiscale approach can significantly reduce the computational cost of the problem while maintaining high accuracy.

Generation a solution to the equations of elasticity theory for a layered strip basing on the principle of compressed mappings

A systematic presentation of the modified classical semi-inverse SaintVenant method as an iterative one is given on the example of generating a solution to the differential equations of elasticity theory for a long layered strip. The firstorder differential equations of the plane problem are reduced to the dimensionless form and replaced by integral equations with respect to the transverse coordinate, just as it is done in the Picard method of simple iterations. In this case, a small parameter appears in the integral equations before the integral sign as a multiplying factor, which is used to ensure convergence of solutions in accordance with the Banach’s principle of compressed mappings. The equations and elasticity relations are converted to a form that enables to calculate the unknowns consecutively, so that the unknowns being calculated in one equation are the inputs for the next equation, and etc. Fulfillment of the boundary conditions at the long edges leads to ordinary differential equations for slowly and rapidly changing singular components of the solution with sixteen effective stiffness coefficients that are defined by integrals from the given ones as a stepped function of Young's moduli for each layer. Integrating of these ordinary differential equations makes it possible to obtain the formulas for all the required unknowns of the problem, including transverse stresses that are not defined in the classical theory of the beam and solutions of the edge effect type, and to fulfill all the boundary conditions for the elasticity theory problem. The solution of three boundary value problems of the strip elasticity theory is provided such as for a two-layer strip with layers of the same thickness and different thicknesses, and a strip with an arbitrary number of layers. Formulas for all unknowns of the problem are obtained.

Application of improved Picard iteration method to simulate unsaturated flow and deformation in deformable porous media

In this paper, we investigate the effectiveness of the cascadic multigrid method applied to improved Picard iteration method for nonlinear problems arising in deforming variably saturated porous media. The finite element method with 6-node triangular elements is applied to discretize the space and obtain nonlinear algebraic equations. Then the nonlinear iterative method is used for iterative solution. Since the classical nonlinear Picard iteration method (PI) can be slow to converge, two improved Picard iteration methods are proposed. One is the improved Picard method with iterations k on the coarse grid and number of multiple grids m (PI-MG(m, k)), and the other is the improved Picard method based on the cascadic multigrid method without parameter k (PI-NMG(m)). Three numerical examples are given to verify the effectiveness of the improved method. Results indicate that the convergence rate of PI is the slowest (10–25 nonlinear iterations per time step), followed by PI-MG(m, k) (3–5 nonlinear iterations), and PI-NMG(m) is the fastest (stable 2 nonlinear iterations). The computational efficiency of PI-NMG(m) is improved by about 90% relative to PI and by about 70% relative to AR-PI. This method can be applied to large-scale numerical calculation of seepage and deformation in unsaturated deforming porous media.

Solution of a two-dimensional parabolic model problem in a degenerate angular domain

In this paper, the boundary value problem of heat conduction in a domain was considered, boundary of which changes with time, as well as there is no the problem solution domain at the initial time, that is, it degenerates into a point. To solve the problem, the method of heat potentials was used, which makes it possible to reduce it to a singular Volterra type integral equations of the second kind. The peculiarity of the obtained integral equation is that it fundamentally differs from the classical Volterra integral equations, since the Picard method is not applicable to it and the corresponding homogeneous integral equation has a nonzero solution.

Efficient and effective algebraic splitting‐based solvers for nonlinear saddle point problems

The incremental Picard Yosida (IPY) method has recently been developed as an iteration for nonlinear saddle point problems that is as effective as Picard but more efficient. By combining ideas from algebraic splitting of linear saddle point solvers with incremental Picard‐type iterations and grad‐div stabilization, IPY improves on the standard Picard method by allowing for easier linear solves at each iteration—but without creating more total nonlinear iterations compared to Picard. This paper extends the IPY methodology by studying it together with Anderson acceleration (AA). We prove that IPY for Navier–Stokes and regularized Bingham fits the recently developed analysis framework for AA, which implies that AA improves the linear convergence rate of IPY by scaling the rate with the gain of the AA optimization problem. Numerical tests illustrate a significant improvement in convergence behavior of IPY methods from AA, for both Navier–Stokes and regularized Bingham.

Study on iterative algorithm of full plate cross-flow and counter-flow printed circuit heat exchanger for fluoride-salt-cooled high-temperature advanced reactor

In the concept of Fluoride-Salt-cooled high-Temperature Advanced Reactor (FuSTAR), the Printed Circuit Heat Exchanger (PCHE) is mainly considered in its supercritical carbon dioxide (S–CO2) Brayton cycle secondary loop. The design of the PCHE is based on the property of working fluid provided by the Brayton cycle system design. Therefore, a fast and iterative PCHE design method is required. In this study, a mathematical model based on Picard's iterative method is proposed. The model can solve full plate temperature field with complex boundary conditions and initial values including cross-flow and counter-flow PCHE in parallel. Based on FuSTAR Brayton cycle design conditions, three sizes of cross-flow and counter-flow PCHEs are simulated respectively by the mathematical model. Compared with Computational Fluid Dynamics (CFD) models, a one and three-dimensional coupling model and a three-dimensional model, the mathematical model reduced the computing time from 1 to 10 h to 1–10 s, and the maximum temperature mean deviation is 18.6% for the smallest size case because of the edge effect and local disturbance, but 8.4% for other size. Taking secondary loop recuperator design as example, the computing speed and accuracy of the mathematical model can meet the design requirements of FuSTAR PCHE.

Solving Nonlinear Elliptic PDEs in 2D and 3D Using Polyharmonic Splines and Low-Degree of Polynomials

In this paper, the improved localized method of approximated particular solutions (ILMAPS) using polyharmonic splines (PHS) together with a low-degree of polynomial basis is used to approximate solutions of various nonlinear elliptic Partial Differential Equations (PDEs). The method is completely meshfree, and it uses a radial basis function (RBF) that has no shape parameters. The discretization process is done through a simple collocation technique on a set of points in the local domain of influence. Resulted system of nonlinear algebraic equations is solved by the Picard method.The performance of the proposed method is tested on various nonlinear elliptical problems, including the Poisson-type problems in 2D and 3D with constant or variable coefficients on rectangular or irregular domains and the Poisson–Boltzmann equation with Dirichlet boundary conditions or mixed boundary conditions. The effect of domain shapes in 2D and 3D, types of boundary conditions, and degrees of PHS, and order of polynomial basis are examined. The performance of the method is compared with other bases such as multiquadrics (MQ) basis functions, and with results reported in the literature (method of particular solutions using polynomials). The numerical experiments suggest that ILMAPS with polyharmonic splines yields considerably superior accuracy than other RBFs as well as other approaches reported in the literature for solving nonlinear elliptic PDEs.