--- title: Trapezium-Rule+DJM Method for Nonlinear VIDEs url: https://www.emergentmind.com/topics/trapezium-rule-daftardar-gejji-jafari-djm-method type: topic --- # Trapezium-Rule+DJM Method for Nonlinear VIDEs The Trapezium-Rule+Daftardar-Gejji-Jafari (DJM) method is a third-order, predictor-corrector-type numerical scheme for solving nonlinear Volterra integro-differential equations (VIDEs) and Volterra delay integro-differential equations (VDIDEs). It combines the classical implicit trapezium rule to discretize integration in both the differential and integral terms, with the Daftardar-Gejji–Jafari decomposition method to explicitly resolve the implicit nonlinear step at each mesh point. This approach yields robust accuracy with moderate computational complexity and broad stability properties for a wide class of nonlinear and delay integro-differential problems [2009.11571], [1604.08863]. ## 1. Problem Class and Mathematical Framework The Trapezium-Rule+DJM method targets equations of the general form \[ \begin{cases} u'(x) = g(x,u(x)) + \displaystyle\int_{x_0}^x K(x,t,u(t-\tau))\,dt\,, & x \in [x_0,X],\\ u(x) = \phi(x), & x \in [x_0-\tau,x_0], \end{cases} \] where $g : [x_0,X] \times \mathbb{R}^n \to \mathbb{R}^n$ and $K : [x_0,X]\times[x_0,X]\times\mathbb{R}^n \to \mathbb{R}^n$ are assumed continuous and satisfy Lipschitz conditions in their functional arguments, and $\phi$ is a prescribed continuous history function for the delay case [2009.11571]. The method, in the absence of delay ($\tau=0$), is likewise applicable to classical nonlinear VIDEs [1604.08863]. The main technical conditions for existence and uniqueness are continuity and global Lipschitz bounds: \[ \|g(x,u_1)-g(x,u_2)\|\le L_1\|u_1-u_2\|, \quad \|K(x,t,v_1)-K(x,t,v_2)\|\le L_2\|v_1-v_2\|. \] Under these assumptions, there exists a unique solution on a local interval about the initial point via Picard–Lindelöf theory [2009.11571], [1604.08863]. ## 2. Discretization via the Implicit Trapezium Rule The interval $[x_0,X]$ is partitioned into $x_j = x_0 + jh$, $j = 0,\dots,N$, with step-size $h = (X-x_0)/N$. Discretization of the differential equation over $[x_j,x_{j+1}]$ proceeds by direct integration followed by trapezoidal quadrature of both the local increment and the nested Volterra integral terms. For the delay case, past approximations $u_{i-M}$ ($M=\tau/h$ for integer $\tau/h$) are utilized. The resulting fully implicit update for the general case is \[ \begin{aligned} u_{j+1} = u_j &+ \frac{h}{2}[g(x_j, u_j) + g(x_{j+1}, u_{j+1})] \\ &+ \frac{h^2}{4}[K(x_j, x_j, u_{j-M}) + K(x_{j+1}, x_{j+1}, u_{j+1-M}) ] \\ &+ \frac{h^2}{2} \sum_{i=1}^j [K(x_j, x_i, u_{i-M}) + K(x_{j+1}, x_i, u_{i-M})] + O(h^3), \end{aligned} \] where $u_j \approx u(x_j)$. The same structure applies without the delay by $u_{i-M} \to u_i$ [2009.11571], [1604.08863]. This formula is implicitly nonlinear in $u_{j+1}$ and (for delay) depends on an extended history of the approximate solution to handle the offset argument of $u$ in $K$. ## 3. Explicit Resolution Using the Daftardar-Gejji-Jafari (DJM) Method The semi-implicit update can be written in the fixed-point form \[ u_{j+1} = M_1 + \frac{h}{2}g(x_{j+1}, u_{j+1}), \] where $M_1$ aggregates all quadrature terms independent of $u_{j+1}$ [2009.11571]. The DJM method resolves this implicit equation by constructing the series \[ u = g_0 + N(u), \text{ with } N(u)=\frac{h}{2}g(x_{j+1}, u), \quad g_0 = M_1. \] The DJM three-term truncation sets \[ \begin{cases} u^{(0)} = g_0, \\ u^{(1)} = N(u^{(0)}), \\ u^{(2)} = N(u^{(0)} + u^{(1)}) - N(u^{(0)}), \\ u_{j+1} \approx u^{(0)} + u^{(1)} + u^{(2)}, \end{cases} \] which simplifies to the compact two-stage update: \[ M_2 = M_1 + \frac{h}{2}g(x_{j+1}, M_1), \quad u_{j+1} = M_1 + \frac{h}{2}g(x_{j+1}, M_2) \] [2009.11571], [1604.08863]. In the non-delay case, similar logic applies but with updated quadrature terms relevant to the convolution structure of the kernel. ## 4. Summary of Algorithmic Steps The following table summarizes the main computational stages per step: | Stage | Key Calculation | Description | |----------------|------------------------------------------------------------------------------------------------|--------------------------------| | Step 1 | $M_1$ by trapezoidal quadrature | Predictor evaluation | | Step 2 | $M_2 = M_1 + \frac{h}{2}g(x_{j+1}, M_1)$ | Predictor-corrector stage | | Step 3 | $u_{j+1} = M_1 + \frac{h}{2}g(x_{j+1}, M_2)$ | Final update | | Step 4 | $j\to j+1$ | Advance mesh index | All historical values required for delayed arguments are drawn from previously computed points, requiring storage of $u_{j-M}$, $u_{j}$, ... as appropriate [2009.11571]. In classical (non-delay) VIDEs, $K$-terms use available $y_i$ directly [1604.08863]. ## 5. Convergence, Error Bounds, and Stability The combined third-order Trapezium+DJM scheme achieves global error \[ \max_{0\leq j \leq N}\|u(x_j) - u_j\| \leq C h^3 \] under the assumed continuity and Lipschitz hypotheses [2009.11571], [1604.08863]. This order follows from a local truncation error $O(h^4)$ and stability established via discrete Grönwall-type estimates. The method, as implemented, is globally third-order accurate for smooth $g$, $K$. Stability and bifurcation analyses have been conducted for representative linear test equations, with stability regions delineated in the $(u,v)$ plane (where $u = h\alpha$, $v=h^2\beta$ in the ODE+integral test cases) [1604.08863]. The method possesses a broad A-stable region and the discrete bifurcation boundaries coincide with continuous ones in the vanishing step-size limit. For the delay case, full characterization of stiff stability is yet to be presented. For non-stiff cases, observed error reduction as $h$ halves is consistent with third-order global accuracy [2009.11571]. ## 6. Numerical Examples and Empirical Assessment Demonstrative computations are provided for both linear and nonlinear VIDEs/VDIDEs: - For $u'(x) = e^{-1}(1-e^x) + u(x) + \int_0^x u(t-1)\,dt$, $u(0)=1$, $u(x<0) = e^x$ (exact $u(x) = e^x$), step-sizes $h=0.01, 0.02, 0.10$ yield maximum absolute errors of $4.6\times10^{-5}$, $1.8\times10^{-4}$, $4.4\times10^{-3}$, confirming third-order convergence [2009.11571]. - For the Day–Wolfe, Dehghan–Salehi, and further nontrivial nonlinear test problems, the method exceeds the accuracy and efficiency of classical third-order Runge–Kutta, multistep, and meshless alternatives, as detailed in comprehensive tabular comparisons [1604.08863]. All analyzed problems confirm the predicted order and efficiency, with CPU times scaling favorably for moderate step-sizes. ## 7. Strengths, Limitations, and Implementation Considerations **Advantages**: - Achieves third-order global accuracy using only trapezoidal and DJM stages, avoiding full nonlinear solves. - Explicit in the correction stage—each step only requires two new $g$ evaluations and modest kernel quadrature. - Well-suited for delay equations via index shifting. - Demonstrated empirical superiority on standard test problems. **Limitations**: - $C^2$-smoothness of $g$ and $K$ is required to maintain $O(h^3)$ error. - Storage of solution histories over $[x_0-\tau, x_0]$ is mandatory; for large delay-to-step-size ratios, this may be substantive. - For stiff kernels or right-hand sides, full implicit or adaptive methods may be needed; stability for such cases remains uncharacterized [2009.11571], [1604.08863]. For very large $N$, more efficient memory management (such as FFT-based history compression) may be beneficial [1604.08863]. Step-size selection can be tuned for error tolerance, and practical convergence assessed via embedded differences between DJM predictor and corrector. In summary, the Trapezium-Rule+Daftardar-Gejji-Jafari method constitutes a rigorously analyzed, high-order, and efficient technique for broad classes of nonlinear VIDEs, with or without delay, leveraging the synergy of classical quadrature and modern decomposition iteration [2009.11571], [1604.08863]. Source: https://www.emergentmind.com/topics/trapezium-rule-daftardar-gejji-jafari-djm-method