Trapezium-Rule+DJM Method for Nonlinear VIDEs

• The Trapezium-Rule+DJM method is a third-order predictor-corrector scheme for solving nonlinear Volterra integro-differential equations with a focus on both delay and non-delay cases.
• It discretizes integration using the implicit trapezium rule and resolves the nonlinear step explicitly through the Daftardar-Gejji-Jafari decomposition method.
• The approach delivers robust accuracy, moderate computational complexity, and broad stability properties as demonstrated in various numerical tests and theoretical analyses.

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 (Jhinga et al., 2020, Bhalekar et al., 2016).

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 (Jhinga et al., 2020). The method, in the absence of delay ($\tau=0$), is likewise applicable to classical nonlinear VIDEs (Bhalekar et al., 2016).

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 (Jhinga et al., 2020, Bhalekar et al., 2016).

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 $g : [x_0,X] \times \mathbb{R}^n \to \mathbb{R}^n$0 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 $g : [x_0,X] \times \mathbb{R}^n \to \mathbb{R}^n$1 ($g : [x_0,X] \times \mathbb{R}^n \to \mathbb{R}^n$2 for integer $g : [x_0,X] \times \mathbb{R}^n \to \mathbb{R}^n$3) are utilized.

The resulting fully implicit update for the general case is

$g : [x_0,X] \times \mathbb{R}^n \to \mathbb{R}^n$4

where $g : [x_0,X] \times \mathbb{R}^n \to \mathbb{R}^n$5. The same structure applies without the delay by $g : [x_0,X] \times \mathbb{R}^n \to \mathbb{R}^n$6 (Jhinga et al., 2020, Bhalekar et al., 2016).

This formula is implicitly nonlinear in $g : [x_0,X] \times \mathbb{R}^n \to \mathbb{R}^n$7 and (for delay) depends on an extended history of the approximate solution to handle the offset argument of $g : [x_0,X] \times \mathbb{R}^n \to \mathbb{R}^n$8 in $g : [x_0,X] \times \mathbb{R}^n \to \mathbb{R}^n$9.

3. Explicit Resolution Using the Daftardar-Gejji-Jafari (DJM) Method

The semi-implicit update can be written in the fixed-point form

$$K : [x_0,X]\times[x_0,X]\times\mathbb{R}^n \to \mathbb{R}^n$$0

where $$K : [x_0,X]\times[x_0,X]\times\mathbb{R}^n \to \mathbb{R}^n$$1 aggregates all quadrature terms independent of $$K : [x_0,X]\times[x_0,X]\times\mathbb{R}^n \to \mathbb{R}^n$$2 (Jhinga et al., 2020). The DJM method resolves this implicit equation by constructing the series

$$K : [x_0,X]\times[x_0,X]\times\mathbb{R}^n \to \mathbb{R}^n$$3

The DJM three-term truncation sets

$$K : [x_0,X]\times[x_0,X]\times\mathbb{R}^n \to \mathbb{R}^n$$4

which simplifies to the compact two-stage update: $$K : [x_0,X]\times[x_0,X]\times\mathbb{R}^n \to \mathbb{R}^n$$5 (Jhinga et al., 2020, Bhalekar et al., 2016).

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	$$K : [x_0,X]\times[x_0,X]\times\mathbb{R}^n \to \mathbb{R}^n$$6 by trapezoidal quadrature	Predictor evaluation
Step 2	$$K : [x_0,X]\times[x_0,X]\times\mathbb{R}^n \to \mathbb{R}^n$$7	Predictor-corrector stage
Step 3	$$K : [x_0,X]\times[x_0,X]\times\mathbb{R}^n \to \mathbb{R}^n$$8	Final update
Step 4	$$K : [x_0,X]\times[x_0,X]\times\mathbb{R}^n \to \mathbb{R}^n$$9	Advance mesh index

All historical values required for delayed arguments are drawn from previously computed points, requiring storage of $\phi$0, $\phi$1, ... as appropriate (Jhinga et al., 2020).

In classical (non-delay) VIDEs, $\phi$2-terms use available $\phi$3 directly (Bhalekar et al., 2016).

5. Convergence, Error Bounds, and Stability

The combined third-order Trapezium+DJM scheme achieves global error

$\phi$4

under the assumed continuity and Lipschitz hypotheses (Jhinga et al., 2020, Bhalekar et al., 2016). This order follows from a local truncation error $\phi$5 and stability established via discrete Grönwall-type estimates. The method, as implemented, is globally third-order accurate for smooth $\phi$6, $\phi$7.

Stability and bifurcation analyses have been conducted for representative linear test equations, with stability regions delineated in the $\phi$8 plane (where $\phi$9, $\tau=0$0 in the ODE+integral test cases) (Bhalekar et al., 2016). 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 $\tau=0$1 halves is consistent with third-order global accuracy (Jhinga et al., 2020).

6. Numerical Examples and Empirical Assessment

Demonstrative computations are provided for both linear and nonlinear VIDEs/VDIDEs:

• For $\tau=0$2, $\tau=0$3, $\tau=0$4 (exact $\tau=0$5), step-sizes $\tau=0$6 yield maximum absolute errors of $\tau=0$7, $\tau=0$8, $\tau=0$9, confirming third-order convergence (Jhinga et al., 2020).
• 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 (Bhalekar et al., 2016).

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 $$\|g(x,u_1)-g(x,u_2)\|\le L_1\|u_1-u_2\|, \quad \|K(x,t,v_1)-K(x,t,v_2)\|\le L_2\|v_1-v_2\|.$$0 evaluations and modest kernel quadrature.
• Well-suited for delay equations via index shifting.
• Demonstrated empirical superiority on standard test problems.

Limitations:

• $$\|g(x,u_1)-g(x,u_2)\|\le L_1\|u_1-u_2\|, \quad \|K(x,t,v_1)-K(x,t,v_2)\|\le L_2\|v_1-v_2\|.$$1-smoothness of $$\|g(x,u_1)-g(x,u_2)\|\le L_1\|u_1-u_2\|, \quad \|K(x,t,v_1)-K(x,t,v_2)\|\le L_2\|v_1-v_2\|.$$2 and $$\|g(x,u_1)-g(x,u_2)\|\le L_1\|u_1-u_2\|, \quad \|K(x,t,v_1)-K(x,t,v_2)\|\le L_2\|v_1-v_2\|.$$3 is required to maintain $$\|g(x,u_1)-g(x,u_2)\|\le L_1\|u_1-u_2\|, \quad \|K(x,t,v_1)-K(x,t,v_2)\|\le L_2\|v_1-v_2\|.$$4 error.
• Storage of solution histories over $$\|g(x,u_1)-g(x,u_2)\|\le L_1\|u_1-u_2\|, \quad \|K(x,t,v_1)-K(x,t,v_2)\|\le L_2\|v_1-v_2\|.$$5 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 (Jhinga et al., 2020, Bhalekar et al., 2016).

For very large $$\|g(x,u_1)-g(x,u_2)\|\le L_1\|u_1-u_2\|, \quad \|K(x,t,v_1)-K(x,t,v_2)\|\le L_2\|v_1-v_2\|.$$6, more efficient memory management (such as FFT-based history compression) may be beneficial (Bhalekar et al., 2016). 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 (Jhinga et al., 2020, Bhalekar et al., 2016).