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By employing the third-order Chebyshev collocation technique along with relevant wavelets, we tackle a third-order singular fractional partial differential equation (PDE). We directly build the Chebyshev operation matrix of the third kind, avoiding the use of the block-pulse function or any approximations. To reduce the order of equation in this approach, we transform the higher-order PDEs into a system of PDEs. Next, we utilize the third-kind Chebyshev wavelet collocation method to convert the resulting system from the prior step into a set of algebraic equations. To demonstrate the method’s effectiveness, we apply it to the time-fractional Camassa–Holm equation and a third-order time-singular PDE. The outcomes are compared with those from several established methods to illustrate the method’s efficiency and practicality.

This paper proposes a new approach to solving three-dimensional linear and nonlinear Volterra integral equations (3D-VIEs) of the second kind using the Taylor collocation method (TCM). We develop an algorithm utilizing Taylor polynomials to approximate solutions of 3D-VIEs within a finite-dimensional space. Moreover, we perform convergence analysis to validate the efficacy of our method. Additionally, we show comparison examples to demonstrate the method’s effectiveness. The comparison highlights the efficiency and reliability of our approach.

This paper presents a novel approach to solve nonlinear second-order Volterra integro-differential equations (VIDEs) using the Iterative Collocation Method. We focus on specific VIDEs with smooth functions, aiming to obtain numerical solutions without the need to solve algebraic equation systems. Our method, based on Lagrange polynomials within a spline space, offers direct implementation, high convergence order, and efficient determination of approximation solution coefficients. Through theoretical analysis and numerical experiments, we illustrate the efficacy and accuracy of our approach, contributing to the improvement of numerical techniques for solving nonlinear VIDEs.

In this paper we explore one of the most important features of the Galerkin method, which is to achieve high accuracy with a relatively modest computational effort, in the dynamics of Robinson–Trautman spacetimes.

In this study, we introduce a new procedure for the numerical solution of linear and nonlinear integral equations by combining the Monte Carlo simulation and collocation method. One of the major advantages of this method is reducing calculations by converting the integral equation to a system of algebraic equations. The numerical examples of various types of integral equations indicate the efficiency and accuracy of the proposed method. The results show this method is able to solve linear and nonlinear integral equations and also nonlinear weakly singular equations, with remarkable accuracy. Finally, the method is generalized for solving integral-differential equation of the neutron transport for the stationary case.

This paper focuses on numerical solutions of time fractional nonlinear Korteweg–de Vries–Burgers equation formulated with Caputo’s fractional derivative. For this purpose, a framework of combinations of collocation method with the finite-element method is provided using trigonometric quintic B-spline basis. The method consists of both spatial discretization and temporal discretization using approximate solution and Crank–Nicolson approach. Discretizing fractional derivative is made using $L1(0<\gamma \le 1)$ algorithm which is derived from the definition of Caputo derivative using an approximate function. The stability analysis is established using von-Neumann stability technique. The numerical results obtained using the collocation method are presented via tables and graphics. The novel results demonstrate the efficiency and reliability of the method.

The steady two-dimensional flow of an incompressible electrically conducting fluid near the stagnation point on a stretching sheet is investigated. The governing highly nonlinear PDEs are altered into highly nonlinear ODE by utilizing non-similar and similar variables and then solved numerically using a Hermite wavelet collocation method (HWCM). It is interesting to note that the results of the previous authors are in good agreement with the results of this study tabulated which is evident from the tabular values. Further, the effects of the magnetic parameter, viscoelastic parameter, stagnation point flow, and stretching of the sheet parameters on the flow field characteristics are obtained and discussed. The Lorentz force opposes the flow. Further, the fluid flow velocity decreases as the viscoelastic parameter increases. This practically indicates that the characteristics of fluid flow in viscoelastic fluids may be influenced by modifying the change in the viscoelastic parameter.

This paper discusses a numerical study of a category of fractional generalized Cattaneo models. Non-Newtonian fluids have been widely used in engineering and industry throughout the last decades. The above model is treated using two autonomous consecutive spectral collocation strategies. For the current model, our technique has proven to be more accurate, efficient, and workable. The analysis indicates that the spectral method is exponentially convergent.

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.

We discuss the basis-spline collocation method for the lattice solution of boundary-value differential equations, drawing particular attention to the difference between lattice and continuous collocation methods. Spectral properties of the basis-spline lattice representation of the first and second spatial derivatives are studied for the case of periodic boundary conditions with homogeneous lattice spacing and compared to spectra obtained using traditional finite-difference schemes. Basis-spline representations are shown to give excellent resolution on small-length scales and to satisfy the chain rule with good fidelity for the lattice-derivative operators using high-order splines. Application to the one-dimensional Dirac equation shows that very high-order spline representations of the Hamiltonian on odd lattices avoid the notorious spectral-doubling problem.

In this research, a hybrid numerical method combining cosine and sine (CAS) wavelets and Green’s function approach is created to acquire the arrangements of fractional Lane–Emden Problem. The suggested methodology for detecting the solution of nonlinear equations dependent on variations of germinal algorithms is applied on nonlinear fractional Lane–Emden Problem under some smooth conditions and results in an iterative scheme of nonlinear equations Because of its efficiency, this technique is applied on a large variety of equations from the boundary value problems to the optimization. This paper is extending the suggested methodology technique for fractional Lane–Emden Problem. Moreover, the feature of the present novel method is utilized to convert the problem under observance into a system of algebraic equations that can be illuminated by suitable algorithms. A rapprochement of results has likewise been obtained using the present strategy and those reported using other techniques seem to indicate the precision and computational efficiency to establish the suitability of the Green-CAS wavelet method.

In this paper, a hybrid approach is presented for the numerical solution of three-dimensional parabolic partial differential equations. This new approach is applicable to both linear and nonlinear parabolic problems including systems. This hybrid numerical technique is based on the Haar wavelet collocation technique and the finite difference method. In this technique, the space derivative is approximated by truncated Haar wavelet series whereas the time derivative is approximated by finite difference method. The aforementioned proposed algorithms are very simple and can easily be implemented in any computer-oriented language efficiently. In order to demonstrate the efficiency and better accuracy of the newly developed numerical technique it is applied to some well-known examples from previous literature that comprises linear and nonlinear three-dimensional parabolic equations including systems. The obtained results affirm better accuracy and widespread applicability of the newly proposed numerical technique for a range of benchmark problems.

This paper deals with the numerical solution of the retarded potential integral equation using a collocation type boundary element method. This method is widely used in practice but often suffers from stability problems. The purpose of the paper is to carry out a stability analysis of the numerical scheme and examine how any instability arises. This paper will then propose a method for overcoming this stability problem. A comparison with an exact solution demonstrates that the approach proposed here is effective for the case of a sphere.

Recently, a conditionally stable explicit time integration scheme using the cubic B-spline function was proposed for solving the structural dynamic problems. This paper presents a new unconditionally stable version of the previous algorithm, based on the uniform cubic B-spline piecewise polynomial approximations and collocation method. First, the method is implemented to solve the differential equation of motion for the single-degree-of-freedom (SDOF) systems. Then, it is generalized for the multi-degree-of-freedom (MDOF) systems. In this paper, a simple step-by-step algorithm is presented for the proposed method, with the stability and accuracy analyses carried out. The unconditional stability of the method is achieved through use of an adjustable collocation parameter $\alpha$. The computational accuracy and efficiency of the proposed method are demonstrated in three numerical examples.

Theoretical modeling of dynamic processes in chemical engineering often implies the numeric solution of one or more partial differential equations. The complexity of such problems is increased when the solutions exhibit sharp moving fronts. An efficient adaptive multiresolution numerical method is described for solving systems of partial differential equations. This method is based on multiresolution analysis and interpolating wavelets, that dynamically adapts the collocation grid so that higher resolution is automatically attributed to domain regions where sharp features are present. Space derivatives were computed in an irregular grid by cubic splines method. The effectiveness of the method is demonstrated with some relevant examples in a chemical engineering context.

The radial basis functions meshfree method is a research method for thin plate problem which has gradually developed into a more mature meshfree method. It includes finite element, radial basis functions meshfree collocation method, etc. In this paper, we introduce the multilevel radial basis function collocation method for the fourth-order thin plate problem. We use nonsymmetric Kansa multilevel radial basis function collocation method to solve the fourth-order thin plate problem. Two numerical examples based on Wendland’s $C^6$ and $C^8$ functions are given to examine that the convergence of the multilevel radial basis function collocation method which is good for solving the fourth-order thin plate problem.

In this study, we develop a generalized wavelet-based collocation method to solve the fractional Pennes bioheat transfer model during hyperthermia treatment. Unlike the existing operational matrix methods based on orthogonal functions, we formulate the Haar wavelet operational matrices of general order integration without using the block pulse functions. Consequently, the governing problem is transformed into an equivalent system of algebraic equations, which can be tackled with any classical method. Some prime features of the proposed method include no requirement of the inverse of the Haar matrices, no need to convert the boundary value problem into the initial-value problem, which in turn eliminates the possibility of unstable solutions. The proposed technique is testified for different values of fractional parameter $\alpha$ and is observed that as the fractional parameter $\alpha$ increases, the tissue temperature at the target region also increases appreciably. Moreover, the obtained results also indicate that the overall time taken to attain the hyperthermia temperature for the fractional model is comparatively less than the classical bioheat model.

In this paper, we develop an efficient wavelet method based on the Fibonacci polynomials for solving the Pennes bioheat transfer equation. The formulation of the proposed technique is started with the construction of Fibonacci wavelets by using Fibonacci polynomials and then applying spectral collocation technique to transform the given problem into a system of an algebraic equation, that can be solved using the Newton method. Some results related to error estimate and convergence analysis of the proposed scheme are also investigated. The applicability and accuracy of the present technique are elucidated by a comparison with the exact solution and those of the other methods found in the recent literature. The obtained results show that the proposed technique is an effective tool for solving Pennes bioheat transfer equations and can also be used for solving similar types of various partial differential equations numerically.

The chemical master equation (CME), formulated from the Markov assumption of stochastic processes, offers an accurate description of general chemical reaction systems. This paper proposes a collocation method using radial basis functions to numerically approximate the solution to the CME. Numerical results for some systems biology problems show that the collocation approximation method has good potential for solving large-scale CMEs.

The main purpose of this work is to provide a numerical approach for linear second-order differential and integro-differential equations with constant delay. An algorithm based on the use of Taylor polynomials is developed to construct a collocation solution $u\in S_m^{(1)}({\mathrm{\Pi }}_N)$ for approximating the solution of second-order linear DDEs and DIDEs. It is shown that this algorithm is convergent. Some numerical examples are included to demonstrate the validity of this algorithm.