Hybrid Block Method (HBM) solvers for initial value problems (IVPs).

HBMIVP is a suite of solvers based on the hybrid block method for solving ordinary differential equations (ODEs) and partial differential equations (PDEs). The implementation includes both Picard and Quasilinearization (QLM) schemes, with options for fixed-step and adaptive-step integration. The solvers are designed to handle stiff and non-stiff systems, as well as higher-order IVPs, in a structured and efficient way.

• Solvers for first, second, and third-order IVPs.
• Picard and Quasilinearization (QLM) methods.
• Fixed-step and adaptive-step implementations.
• Support for stiff and non-stiff problems.
• Examples ranging from simple test equations to PDEs (e.g., the heat equation).
• High-accuracy results validated against analytical solutions and MATLAB’s ode45.

A typical workflow:

% Define problem

f = @(t,y) -100y + 99sin(t); % RHS of ODE

y0 = [1; 11]; % Initial conditions

% Solver options

opts = struct('method','qlm','h',1e-2,'M',4);

% Call solver

[t, y] = hbmivp(f, [0, 1], y0, opts);

% Compare with analytical solution

y_exact = cos(10t) + sin(10t) + sin(t);

plot(t, y(:,1), 'b-', t, y_exact, 'r--')

legend('HBM solution', 'Exact solution')

The examples folder contains:

• Examples 1 to 7: Test problems for first- to third-order IVPs.
• Examples 8 and 9: PDEs solved using spectral discretization in space and HBM in time. Comparison with MATLAB’s ode45 and ode15s is included for examples.

Contributions are welcome! Please open issues or submit pull requests to:

• Add new examples
• Improve solver efficiency
• Extend to fractional or stochastic IVPs/PDEs

If you use HBMIVP in your research, please cite:

@misc{hbmivp2025,

author = {Shina D Oloniiju},

title = {HBMIVP: Hybrid block method solvers for IVPs},

year = {2025},

howpublished = {\url{https://github.com/Shinadan/HBMSolvers}}

}