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This study explores the Petrov–Galerkin method’s application in solving a linear fourth-order ordinary beam equation of the form u″″+qu=fu^{\prime\prime} ^{\prime\prime} +qu=f. The equation entails two distinct boundary conditions: pinned–pinned conditions on uu and u′u^{\prime} , and clamped–clamped conditions on uu and u″{u}^{^{\prime\prime} }. To satisfy these boundary conditions, we have built two sets of basis functions. The explicit forms of all spectral matrices were reported. The nonhomogeneous boundary conditions were easily handled using perfect transformations, ensuring the numerical solution’s accuracy. Detailed analysis of the method’s convergence was studied. Some numerical examples were presented, accompanied by comparisons with other existing methods in the literature.

In this research, a compact combination of Chebyshev polynomials is created and used as a spatial basis for the time fractional fourth-order Euler–Bernoulli pinned–pinned beam. The method is based on applying the Petrov–Galerkin procedure to discretize the differential problem into a system of linear algebraic equations with unknown expansion coefficients. Using the efficient Gaussian elimination procedure, we solve the obtained system of equations with matrices of a particular pattern. The L∞{L}_{\infty } and L2{L}_{2} norms estimate the error bound. Three numerical examples were exhibited to verify the theoretical analysis and efficiency of the newly developed algorithm.

The current article introduces a Petrov–Galerkin method (PGM) to address the fourth-order uniform Euler–Bernoulli pinned–pinned beam equation. Utilizing a suitable combination of second-kind Chebyshev polynomials as a basis in spatial variables, the proposed method elegantly and simultaneously satisfies pinned–pinned and clamped–clamped boundary conditions. To make PGM application easier, explicit formulas for the inner product between these basis functions and their derivatives with second-kind Chebyshev polynomials are derived. This leads to a simplified system of algebraic equations with a recognizable pattern that facilitates effective inversion to produce an approximate spectral solution. Presentations are made regarding the method's convergence analysis and the computational cost of matrix inversion. The efficiency of the method described in precisely solving the Euler–Bernoulli beam equation under different scenarios has been validated by numerical testing. Additionally, the procedure proposed in this paper is more effective compared to other existing techniques. [ABSTRACT FROM AUTHOR]

In this study, we utilize a specific combination of orthogonal Gegenbauer polynomials as basis functions that satisfy homogeneous boundary conditions in the spatial variable and homogeneous initial conditions in the temporal variable. We derive an expression for the fractional derivative of the temporal basis and another expression for the second‐order derivative of the spatial basis. Subsequently, we discretize the nonlinear Fisher problem using the spectral collocation method to transform the problem into the solution of a nonlinear system of algebraic equations. We then solve this resulting system using Newton's method with an initial guess approaching zero. Furthermore, we conduct an error analysis of the proposed method and assess its applicability through three test problems, providing comparisons for validation. [ABSTRACT FROM AUTHOR]

This paper presents a novel numerical spectral scheme to handle the time-fractional KdV–Burgers' equation, which is very important in both physics and engineering. The scheme basically uses the tau approach combined with Gegenbauer polynomials to provide accurate and stable numerical solutions. Instead of solving the differential problem together with the conditions, we solve a system of algebraic equations. The method can handle complex boundary conditions. Some numerical experiments are exhibited to demonstrate that this approach is highly efficient and produces results that are better than some existing numerical methods in the literature. This technique offers more advanced solutions for time-fractional problems in various fields. [ABSTRACT FROM AUTHOR]

A novel Haar scale-3 wavelet collocation technique is proposed in this study for dealing with a specific type of parabolic Buckmaster second-order nonlinear partial differential equation in a dispersive system and Chafee-Infante second-order non-linear partial differential equation (PDE) in a solitary system. Using Haar scale-3 (HSW-3) wavelets, the system approximates the space and time derivatives. To develop both an implicit and explicit analytical model for the dispersive and solitary system, the collocation approach is employed in conjunction with the discretization of space and time variables. We have examined the effectiveness, applicability, and veracity of the proposed computational approach using a variety of numerical problems with nonlinearity and numerous significant source terms. Additionally, the outcomes are graphically presented and organized. We achieved accuracy with the proposed methods even with a small selection of collocation locations. [ABSTRACT FROM AUTHOR]

Herein, we present an algorithm for handling fractional delay differential equations (FDDEs). Chebyshev polynomials (CPs) class of the seventh kind is a subclass of the generalized Gegenbauer (ultraspherical) polynomials. The members of this class make up the basis functions in this paper. Our suggested numerical algorithm is derived using new theoretical findings about these polynomials and their shifted counterparts. We will use the Tau method to convert the FDDE with the governing conditions into a linear algebraic system, which can then be solved numerically using a suitable procedure. We will give a detailed discussion of the convergence and error analysis of the shifted Chebyshev expansion. Lastly, some numerical examples are provided to verify the precision and applicability of the proposed strategy. [ABSTRACT FROM AUTHOR]

Let J be a symmetric Jacobi matrix associated with some Toda lattice. We find conditions for Jacobi matrix J to admit factorization J = LU (or J = il£) with L (or £) and U ( or il) being lower and upper triangular two-diagonal matrices, respectively. In this case, the Darboux transformation of J is the symmetric Jacobi matrix J^' = UL (or = £il), which is associated with another Toda lattice. In addition, we found explicit transformation formulas for orthogonal polynomials, m-functions and Toda lattices associated with the Jacobi matrices and their Darboux transformations. [ABSTRACT FROM AUTHOR]

This study deals with a fitted third-degree B-spline collocation method for two parametric singularly perturbed parabolic problems with a time lag. The proposed method comprises the Cranck-Nicolson method for time discretization and the third-degree B-spline method spatial variable discretization. Rigorous numerical experimentations were carried out on some test examples. The obtained numerical results depict that the proposed scheme is more accurate than some methods existing in the literature. Parameter convergence analysis of the scheme is carried out and shows the present scheme is (ε - μ)-uniform convergent with the order of convergence ((Δt)² + ℓ²). [ABSTRACT FROM AUTHOR]

The main goal of this research work is to provide a numerical technique based on choosing a set of basis functions for handling the third-order time-fractional Korteweg–De Vries Burgers’ equation. The trial functions are selected for the shifted second-kind Chebyshev polynomials (S2KCPs) compatible with the problem’s governing initial and boundary conditions. The spectral tau method transforms the equation and its underlying conditions into a nonlinear system of algebraic equations that can be efficiently numerically inverted with the standard Newton’s iterative procedures after the approximate solutions have been expressed as a double expansion of the two chosen basis functions. The truncation error is estimated. Various numerical examples are displayed together with comparisons to other approaches in the literature to show the applicability and accuracy of the provided methodology. Different numerical models are displayed and compared to other methods in the literature. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction of certain polynomials related to Lucas polynomials, namely, modified Lucas polynomials. The constructed modified Lucas polynomials are utilized as basis functions for the numerical treatment of the linear and non-linear second-order boundary value problems (BVPs) involving some specific important problems such as singular and Bratu-type equations. To derive our proposed algorithms, the operational matrix of derivatives of the modified Lucas polynomials is established by expressing the first-order derivative of these polynomials in terms of their original ones. The convergence analysis of the modified Lucas polynomials is deeply discussed by establishing some inequalities concerned with these modified polynomials. Some numerical experiments accompanied by comparisons with some other articles in the literature are presented to demonstrate the applicability and accuracy of the presented algorithms. [ABSTRACT FROM AUTHOR]