Polynomial Collocation Method Research Articles

On the solution of MHD Jeffery–Hamel problem involving flow between two nonparallel plates with a blood flow application

AbstractThe Jeffery–Hamel flow phenomenon appears in a variety of real‐world applications involving the flow of two nonparallel plates. BY using a similarity transformation derived from the equation of continuity, partial differential equations determining flow characteristics are translated into nonlinear ordinary differential equations. The problem involves the flow of a specific type of fluid, namely, an incompressible and electrically conducting fluid, between two nonparallel plates. The flow is assumed to be steady, two‐dimensional, and subject to certain boundary conditions. Specifically, the plates are impermeable, and the fluid adheres to a no‐slip condition, resulting in zero fluid velocity at the plates' surfaces. Moreover, the problem incorporates the effects of magnetic fields and pressure fluctuations, making it highly applicable to scenarios, such as blood flow through arteries in the human body, which can be modeled as a special case of the magnetohydrodynamic (MHD) Jeffery–Hamel problem referred to as the (MHD) blood pressure equation. This work compares two numerical approaches for solving the MHDs Jeffery–Hamel problem: B‐spline and Bernstein polynomial collocation. The given approaches are used to discretize and transform the equation into a system of algebraic equations. Matrix algebra techniques are then used to solve the resultant system. A complete error analysis and convergence rates for different grid sizes are derived for both methods and are used to compare the accuracy and efficiency of the two approaches. Both approaches produce correct solutions, according to the numerical findings, although the Bernstein polynomial collocation method is more efficient and accurate than the B‐spline collocation.

A numerical investigation of a well-known nonlinear Newell-Whitehead-Segel equation using the rank polynomial of the star graph

Mathematical models of pattern formation are indispensable tools in various fields, from developmental biology to ecology, providing insights into complex phenomena and contributing to our understanding of the natural world. These patterns have been extensively studied using reaction-diffusion and NewellWhiteheadSegel models. This article intended to find an approximate numerical solution to the NewellWhiteheadSegel equation. The appearance of stripe patterns in two-dimensional systems is explained in nonlinear systems using the NewellWhiteheadSegel equation. Based on the function basis of rank polynomials of star graphs and the well-posed operational matrices, the rank polynomial collocation method is constructed. The alleged rank polynomial collocation method created a system of nonlinear algebraic equations from the nonlinear NewellWhiteheadSegel equation. The nonlinear NewellWhiteheadSegel equation solution is approximated by solving the resulting system via Newton’s Raphson method. Numerical instances are provided to illustrate the validity and effectiveness of the technique. Verification of accuracy is accomplished by calculating error norms. The obtained numerical results show a reasonable degree of consistency with the findings reported in the current literature. The scheme’s primary benefit is the algorithm’s ease of implementation.

A new graph theoretic analytical method for nonlinear distributed order fractional ordinary differential equations by clique polynomial of cocktail party graph

AbstractIn this paper, we presented a new analytical method for one of the rapidly emerging branches of fractional calculus, the distributed order fractional differential equations (DFDE). Due to its significant applications in modeling complex physical systems, researchers have shown profound interest in developing various analytical and numerical methods to study DFDEs. With this motivation, we proposed an easy computational technique with the help of graph theoretic polynomials from algebraic graph theory for nonlinear distributed order fractional ordinary differential equations (NDFODE). In the method, we used clique polynomials of the cocktail party graph as an approximation solution. With operational integration and fractional differentiation in the Caputo sense, the NDFODEs transformed into a system of algebraic equations and then solved by Newton–Raphson's method to determine the unknowns in the Clique polynomial approximation. The proficiency of the proposed Clique polynomial collocation method (CCM) is illustrated with four numerical examples. The convergence and error analysis are discussed in tabular and graphical depictions by comparing the CCM results with the results of existing numerical methods.

Dynamics of pollutant dispersion and solid–fluid interfacial layer in Jeffrey nanofluid flow subjected to waste discharge concentration: Implementation of probabilists’ Hermite polynomial collocation method

The current study explores the flow of a Jeffrey nanofluid with the impact of a non-uniform heat source/sink and magnetic effects over a stretching sheet by accounting for a pollutant’s transient dispersion after being released from a source outside the system. The need to enhance the efficiency of prospective future technologies, including heat exchangers, solar collectors with nanoscale solid particles suspended in the base fluid, powered engines, pharmaceutical procedures, and hybrid microelectronics, has grown recently. Graphene is treated as a nanoparticle and NaA lg as a base liquid. To emphasize the thermal integrity of the flow that is now taken into consideration, the influence of the diameter of the nanoparticle and the liquid–solid interfacial layer is also shown at the molecular level. The modeled equations are transformed into ordinary differential equations by means of the appropriate similarity transformations. Probabilists’ Hermite polynomial collocation technique is employed to solve these nonlinear dimensionless ODEs, and the graphical representations are created for certain significant values of the underlying physical parameters in the flow model. The result reveals that as Deborah number increases, velocity upsurges but temperature declines. A rise in concentration is observed when the local pollutant external source variation parameter rises. The temperature is enhanced with the advanced values of non-uniform heat source/sink parameters.

Application of Polynomial Collocation Method Based on Successive Integration Technique for Solving Delay Differential Equation in Parkinson’s Disease

Application of Polynomial Collocation Method Based on Successive Integration Technique for Solving Delay Differential Equation in Parkinson’s Disease

Study of numerical treatment of functional first-kind Volterra integral equations

<abstract><p>First-kind Volterra integral equations have ill-posed nature in comparison to the second-kind of these equations such that a measure of ill-posedness can be described by $ \nu $-smoothing of the integral operator. A comprehensive study of the convergence and super-convergence properties of the piecewise polynomial collocation method for the second-kind Volterra integral equations (VIEs) with constant delay has been given in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. However, convergence analysis of the collocation method for first-kind delay VIEs appears to be a research problem. Here, we investigated the convergence of the collocation solution as a research problem for a first-kind VIE with constant delay. Three test problems have been fairly well-studied for the sake of verifying theoretical achievements in practice.</p></abstract>

Approximate Solution of a Class of Highly Oscillatory Integral Equations Using an Exponential Fitting Collocation Method

This paper deals with the numerical solution of a class of highly oscillatory Volterra integral equations by collocation methods based on the exponential fitting technique. By reviewing the oscillatory structures of solutions of these problems, we construct an exponential fitting collocation method which is best tuned to capture the qualitative behaviour of the solution of these equations. We also investigate the convergence properties of the proposed collocation solution based on the interpolation remainder. Some numerical examples are provided which illustrate the efficiency and accuracy of the proposed method and confirm its superiority over the polynomial collocation methods.

Collocation-Based Approximation for a Time-Fractional Sub-Diffusion Model

We consider the numerical solution of a one-dimensional time-fractional diffusion problem, where the order of the Caputo time derivative belongs to (0, 1). Using the technique of the method of lines, we first develop from the original problem a system of fractional ordinary differential equations. Using an integral equation reformulation of this system, we study the regularity properties of the exact solution of the system of fractional differential equations and construct a piecewise polynomial collocation method to solve it numerically. We also investigate the convergence and the convergence order of the proposed method. To conclude, we present the results of some numerical experiments.

A new graph-theoretic approach for the study of the surface of a thin sheet of a viscous liquid model

In this study, we considered the model of the surface of a thin sheet of viscous liquid which is known as the Buckmaster equation (BME), and presented a new graph-theoretic polynomial collocation method named the Hosoya polynomial collocation method (HPCM) for the solution of nonlinear Buckmaster equation. In the literature, the majority of the developed numerical methods considered small time step sizes like 0.01s and 0.05s to obtain relatively accurate approximations for the nonlinear BME. This study focused on optimizing the time step sizes by adopting bigger time steps sizes like 1.0s,3.0s, and 5.0s, etc without adversely affecting accuracy. First, using the Gram- Schmidt process, we generated the orthonormal functions from the Hosoya polynomial of the path graph. Then developed the functional integration matrix using orthonormal Hosoya polynomials of path graphs. With this active matrix-involved method, the nonlinear BMEs are transformed into a system of nonlinear equations and solved the equations by Newton’s method through the Mathematica software for unknown coefficients. The exactness of the proposed strategy is tested with two numerical examples. The acquired results contrasted with the current analytical solutions to these problems. Also provided the convergence analysis, comparison of error norms, graphical plots of correlation of HPCM results, and the results of other numerical methods in the literature to validate the productivity and accuracy of the newly developed HPCM.

A novel analytical method for the multi-delay fractional differential equations through the matrix of clique polynomials of the cocktail party graph

This paper provides a unique graph matrix approach based on the Clique polynomials of the Cocktail party graph to effectively solve multi-delay fractional differential equations (MDFDE) with variable coefficients. Clique polynomials of the cocktail party graph’s basis are used as a trial function for the solution of MDFDE. An operational matrix of Caputo fractional derivative of polynomial approximation is employed to transform the MDFDEs into a system of algebraic equations. The unknowns in the system are determined by Newton’s method in Mathematica. The productivity of the suggested strategy is examined with five numerical examples. From all the discussed examples, it is observed that the proposed Clique polynomial collocation method (CPCM) has excellent compatibility with linear MDFDEs by attaining an exact solution. To substantiate the validity of the recommended method, obtained results with comparisons, computation time, convergence analysis, error estimations, and graphical illustrations are offered.

Quasilinear bessel polynomial collocation method for the solutions of nonlinear singular boundary value problems

The Lane–Emden–Fowler-type equations along with boundary conditions form nonlinear singular boundary value problems. Analytical solutions to this type of problem are not easy, which draws the attention of the researchers toward their numerical solutions. Many methods have been used to provide numerical solutions to this type of problem. In this study, we have proposed the quasilinear Bessel polynomial collocation method (QBPCM) for the solution of these problems. The accuracy and efficiency of the QBPCM have been demonstrated by solving several nonlinear singular models that arise in real-life problems. It has been shown that the numerical results obtained by the proposed method have excellent agreement with the exact solutions and better accuracy than other methods.

Numerical solution of Advection–Diffusion Equation using Graph theoretic polynomial collocation method

Water is one of the main constituents on earth for a living. The Advection Diffusion Equation (ADE) serves as an essential water standard model in environmental engineering since water pollution seriously threatens all life. Hence, the study of ADE has become an increasing concern for researchers in recent years. In the present work, we propose a new numerical method named the Hosoya polynomial collocation method (HPCM) for solving one-dimensional linear ADE. In this work, we developed an operational matrix of Integration of the ortho-normal Hosoya polynomials of path graphs (OHPG) to compute the solute concentration numerically by converting ADE into an algebraic system of equations and solving this system of equations using Newton’s method. With the appropriate collocation approach, we developed a graph theoretic polynomial-based solution to the considered ADE. We presented the convergence analysis, comparisons of HPCM findings with ADE’s analytical solutions, comparisons of error norms, and graphical representations of the numerical conclusions to illustrate the productivity of the HPCM. Comparing numerical results with the outcomes of other current numerical methods ensures the validity of the suggested method’s accuracy and efficacy.

COLLOCATION BASED APPROXIMATIONS FOR A CLASS OF FRACTIONAL BOUNDARY VALUE PROBLEMS

A boundary value problem for fractional integro-differential equations with weakly singular kernels is considered. The problem is reformulated as an integral equation of the second kind with respect to, the Caputo fractional derivative of y of order α, with 1 < α < 2, where y is the solution of the original problem. Using this reformulation, the regularity properties of both y and its Caputo derivative z are studied. Based on this information a piecewise polynomial collocation method is developed for finding an approximate solution zN of the reformulated problem. Using zN, an approximation yN for y is constructed and a detailed convergence analysis of the proposed method is given. In particular, the attainable order of convergence of the proposed method for appropriate values of grid and collocation parameters is established. To illustrate the performance of our approach, results of some numerical experiments are presented.

On the convergence of piecewise polynomial collocation methods for variable-order space-fractional diffusion equations

In this study, we present a piecewise polynomial collocation method for the boundary-value problem of variable-order linear space-fractional diffusion equations. The proposed model is transformed to a weakly singular Volterra integral equation (VIE) of the second kind by an auxiliary variable, then a collocation method is constructed and analyzed for the obtained VIE. We demonstrate the existence and uniqueness of the collocation solution, as well as the optimal convergence order of the collocation method. Some numerical experiments are given to illustrate the theoretical results.

Discontinuous piecewise polynomial collocation methods for integral-algebraic equations of Hessenberg type

Discontinuous piecewise polynomial collocation methods for integral-algebraic equations of Hessenberg type

Numerical Approximation of Nonlinear Duffing Oscillator Using a Coupled Approach

Duffing equation can describe many important nonlinear physical systems. In this paper a coupled approach based on quasilinearization and Bessel polynomial collocation method has been suggested to solve nonlinear duffing oscillator equation. The nonlinearity in duffing oscillator can be of various variety. This approach is very efficient, stable and reliable to deal with any kind of nonlinearity. Numerical examples demonstrate the validity and applicability of the approach on various types of nonlinear duffing oscillator equation.

Discontinuous Galerkin approximations to second-kind Volterra integral equations with weakly singular kernel

The discontinuous Galerkin (DG) method is employed to solve second-kind Volterra integral equations (VIEs) with weakly singular kernels. It is proved that the quadrature DG (QDG) method obtained from the DG method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the piecewise discontinuous polynomial collocation method. The convergence theory is established for VIEs for both uniform and graded meshes. Some numerical experiments are given to illustrate the obtained theoretical results.

Analysis of collocation methods for a class of third-kind auto-convolution Volterra integral equations

In this paper, we deal with piecewise polynomial collocation methods for a class of third-kind auto-convolution Volterra integral equations (AVIEs). A weighted exponential norm is introduced to prove the existence and uniqueness of the analytic solution, and the regularity is discussed based on the theory for linear cordial Volterra integral equations (CVIEs). The solvability of collocation methods with uniform meshes is discussed following the step-by-step approach, and the uniform boundedness of collocation solutions is proved by introducing an analogous discrete weighted exponential norm. The optimal convergence property is obtained by some new techniques. Some numerical experiments are given to illustrate the theoretical results.

Study of Interval Type-2 Fuzzy Singular Integro-Differential Equation by Using Collocation Method in Weighted Space

This paper develops the numerical solution of interval type-2 fuzzy singular integro-differential equation. The theory of polynomial collocation method has been introduced to find the numerical solution of interval type-2 fuzzy singular integro-differential equation in different weighted spaces. The equation has been presented in operator form. It has been shown that the operators are bounded which are required for the convergence of the proposed method. Theorems and lemmas have been developed to show the convergence of our proposed method in type-2 fuzzy environment. A numerical algorithm for the collocation method has been presented in the numerical section. A numerical example has been examined to show the validation of our proposed method. Different types of error analysis have been examined in the form of different types of tables and figures. Also, the comparison of different types of error analysis has been examined in different weighted space.

A Space-Time Polynomial Collocation Method for Solving 3D Burgers Equations

Burgers’ equation is a kind of nonlinear partial differential equation which is widely used in various fields. A new space-time polynomial particular solutions collocation method is presented to solve three-dimensional Burgers’ equation. The basic process is divided into two steps. The first step is to find the polynomial particular solutions of the linear differential operator terms (including the time differential term) in the govern equation. The second step is to solve the nonlinear term of three-dimensional Burgers’ equation iteratively. Our proposed method is simple and easy to program. The approximate solution is very accurate. Especially, the stability of the method is excellent, which improves students’ programming ability and deeps their understanding of high-dimensional Burgers’ equation and its practical application.