This article provides a detailed exploration of the SIR epidemic model, starting with its meticulous formulation. The study employs a novel approach called the upper and lower bounds technique to approximate the solution to the SIR model, providing insights into the dynamic interplay between susceptible S, infected I, and recovered R populations. A new parametric solution to this model has been presented. Applying the Adomian decomposition method (ADM) allows for the attaining of highly accurate approximate solutions in the context of the SIR epidemic model. To validate the accuracy and robustness of the proposed approach, a numerical exploration is conducted, considering a diverse range of experimental parameters. This numerical analysis provides valuable insights into the sensitivity and responsiveness of the SIR epidemic model under varying conditions, contributing to the broader understanding of infectious disease dynamics. The interplay between theoretical formulation and numerical exploration establishes a comprehensive framework for studying the SIR model, with implications for refining our ability to predict and manage the spread of infectious diseases.

An explicit approximate solution is obtained for the extended Blasius equation subject to its well-known classical boundary conditions, where the viscosity coefficient is assumed to be positive and temperature-dependent, which arises in several important boundary layer problems in fluid dynamics. This problem extends a previous problem by Cortell (Appl Math Comput 170:706–710, 2005) when the viscosity is constant, in which a numerical solution was obtained. A comparison with other numerical solutions demonstrates that our approximate solution shows an enhancement over some of the existing numerical techniques. Moreover, highly accurate estimates for the skin-friction were calculated and found to be in good agreement with the numerical values obtained by Howarth (Proc R Soc A: Math Phys Eng Sci 164(919):547–579, 1938), Töpfer (Z Math Phys 60:397–398, 1912), and Cortell [34] when the viscosity is equal to 1, and when it is equal to 2.

The purpose of this paper is to establish an existence and uniqueness theorem for the nonlocal fourth-order nonlinear beam differential equations with a parameter \begin{document}$ \begin{equation*} u^{(4)}+A(x)u = \lambda f (x, \ u, \ u''), \ 0<x<1 \end{equation*} $\end{document} subject to the integral boundary conditions: $ \begin{equation*} u(0) = u(1) = \int_{0}^{1}p(x)u(x)dx, \ u''(0) = u''(1) = \int_{0}^{1}q(x)u''(x)dx, \end{equation*} $ where $ A\in \mathbb{C}[0, 1], $ $ \lambda > 0 $ is a parameter and $ p, q \in \mathbb{L}^{1}[0, 1]. $

Solutions of similarity-type for a nonlinear non-classical Stefan problem with temperature-dependent thermal conductivity and a Robin boundary condition are obtained. The analysis of several particular cases are given when the thermal conductivity $ L(f) $ and specific heat $ N(f) $ are linear in temperature such that $ L(f) = \alpha +\delta f $ with $ N(f) = \beta+\gamma f. $ Existence of a similarity type solution also obtained for the general problem by proving the lower and upper bounds of the solution.

Approximate solutions are obtained in implicit forms for the following general form of the nonlinear Stefan problem $\frac{d}{dx}\left[(1+{\delta }_1{y}^p)\frac{dy}{dx}\right]+2x(1+{\delta }_2{y}^p)\frac{dy}{dx}=\frac{4}{Ste}\beta \left(x\right),0<x<\lambda ,$ with $y\left(0\right)=1,y\left(\lambda \right)=0,$ where $\lambda >0$ is a solution to the nonlinear equation $y^{\prime }\left(\lambda \right)=-\frac{2\lambda }{Ste}$, where ${\delta }_i>-1,i=1,2,$$p>0,$ and $Ste$ is the Stefan number, which represents a phase-change problem with a nonlinear temperature-dependent thermal parameters (i.e., thermal conductivity and specific heat) on $(0,\lambda )$.

The purpose of this paper is to establish an existence theorem for a system of nonlinear fourth-order differential equations with two parameters $ \begin{eqnarray*} \left\{ \begin{array}{rcl} u^{(4)}+A(x)u& = &\lambda f(x, u, v, u'', v''), \ 0<x<1, \\ v^{(4)}+B(x)v& = &\mu g(x, u, v, u'', v''), \ 0<x<1 \end{array} \right. \end{eqnarray*} $ subject to the coupled integral boundary conditions: $ \begin{eqnarray*} \left\{ \begin{array}{rcl} u(0) = u'(1) = u'''(1) = 0, \ u''(0)& = & \int_{0}^{1}p(x)v''(x)dx, \\ v(0) = v'(1) = v'''(1) = 0, \ v''(0)& = & \int_{0}^{1}q(x)u''(x)dx, \end{array} \right. \end{eqnarray*} $ where $ A, \ B \in C[0, 1], $ $ p, q\in L^{1}[0, 1], $ $ \lambda > 0, \mu > 0 $ are two parameters and $ f, g: [0, 1]\times[0, \infty)\times[0, \infty)\times(-\infty, 0)\times(-\infty, 0) \rightarrow \mathbb{R} $ are two continuous functions satisfy the growth conditions.